Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity

Panagiotis Paraschis; Georgios E. Zouraris

doi:10.1515/cmam-2022-0217

Open Access Published by De Gruyter March 31, 2023

Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity

Panagiotis Paraschis and Georgios E. Zouraris

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2022-0217

Abstract

We formulate an initial and Dirichlet boundary value problem for a semilinear heat equation with logarithmic nonlinearity over a two-dimensional rectangular domain. We approximate its solution by employing the standard second-order finite difference method for space discretization, and a linearized backward Euler method, or, a linearized BDF2 method for time stepping. For the linearized backward Euler finite difference method, we derive an almost optimal order error estimate in the discrete L t ∞ ⁢ ( L x ∞ ) -norm without imposing mesh conditions, and for the linearized BDF2 finite difference method, we establish an almost optimal order error estimate in the discrete L t ∞ ⁢ ( H x 1 ) -norm, allowing a mild mesh condition to be satisfied. Finally, we show the efficiency of the numerical methods proposed, by exposing results from numerical experiments. It is the first time in the literature where numerical methods for the approximation of the solution to the heat equation with logarithmic nonlinearity are applied and analysed.

Keywords: Heat Equation; Logarithmic Nonlinearity; Dirichlet Boundary Conditions; Finite Difference Discretization; Implicit-Explicit Time-Stepping; Linearized Backward Euler; Linearized BDF2; Optimal Order Error Estimates

MSC 2010: 65M12; 65M06; 65M15

1 Introduction

Let T > 0 , D := [ a 1 , a 2 ] × [ b 1 , b 2 ] ⊂ R 2 , Q := [ 0 , T ] × D , and let u : Q → R be the solution of the following initial and boundary value problem:

(1.1) u t = Δ ⁢ u + g ⁢ ( u ) + f on ⁢ ( 0 , T ] × int ⁡ ( D ) ,

(1.2) u ⁢ ( t , x ) = 0 for all ⁢ ( t , x ) ∈ ( 0 , T ] × ∂ D ,

(1.3) u ⁢ ( 0 , x ) = u 0 ⁢ ( x ) for all ⁢ x ∈ int ⁡ ( D ) ,

where f ∈ C ⁢ ( Q ) , u 0 ∈ C ⁢ ( D ) with

(1.4) u 0 | ∂ D = 0

and g ∈ C ⁢ ( R ) is an odd function given by

(1.5) g ⁢ ( s ) = { 0 , s = 0 , s ⁢ ln ⁡ ( | s | ) , s ≠ 0 , for all ⁢ s ∈ R ,

which can be considered as a simplified version of the Flory–Huggins free energy (cf. [9, 11]). It is easily seen that 𝑔 is not differentiable at zero, and it is not locally Lipschitz over any interval containing zero. However, 𝑔 satisfies the following local one-sided Lipschitz condition (cf. [1]).

Lemma 1.1

For c > e , it holds that

(1.6) ( g ⁢ ( x ) − g ⁢ ( y ) ) ⁢ ( x − y ) ≤ ( 1 + ln ⁡ ( c ) ) ⁢ | x − y | 2 for all ⁢ x , y ∈ [ − c , c ] .

Proof

First, we observe that (1.6) holds trivially when x ⁢ y = 0 . Assuming that x ⁢ y ≠ 0 , we have

(1.7) ( g ⁢ ( x ) − g ⁢ ( y ) ) ⁢ ( x − y ) = ( x − y ) 2 ⁢ ln ⁡ ( | x | ) + y ⁢ ( x − y ) ⁢ ( ln ⁡ ( | x | ) − ln ⁡ ( | y | ) ) ≤ ( x − y ) 2 ⁢ ln ⁡ ( c ) + | y | ⁢ | x − y | ⁢ | ln ⁡ ( | x | ) − ln ⁡ ( | y | ) | .

When | x | = | y | , then (1.6) follows easily from (1.7). When | y | < | x | , then there exists ξ ∈ ( | y | , | x | ) such that ln ⁡ ( | x | ) − ln ⁡ ( | y | ) = 1 ξ ⁢ ( | x | − | y | ) , which, along with (1.7), establishes (1.6) as follows:

( g ⁢ ( x ) − g ⁢ ( y ) ) ⁢ ( x − y ) ≤ ( x − y ) 2 ⁢ ln ⁡ ( c ) + | y | ξ ⁢ | x − y | ⁢ | | x | − | y | | ≤ | x − y | 2 ⁢ ( | y | ξ + ln ⁡ ( c ) ) ≤ | x − y | 2 ⁢ ( 1 + ln ⁡ ( c ) ) .

Since (1.6) is symmetric with respect to 𝑥 and 𝑦, it holds also when | y | > | x | . ∎

The result of Lemma 1.1 could be used to ensure the uniqueness of the solution to the initial and boundary value problem formulated above, proceeding as follows.

Lemma 1.2

The solution to the initial and boundary value problem (1.1)–(1.5) is unique.

Proof

Let 𝑢 and 𝑤 be solutions to problem (1.1)–(1.5), c > max ⁡ { e , max Q ⁡ | u | , max Q ⁡ | w | } and ζ = u − w . Then we have

(1.8) ζ t = Δ ⁢ ζ + ( g ⁢ ( u ) − g ⁢ ( w ) ) on ⁢ ( 0 , T ] × int ⁡ ( D ) .

Taking the L 2 ⁢ ( D ) -inner product of both sides of (1.8) by 𝜁, setting ν ⁢ ( t ) = ∫ D ζ 2 ⁢ ( t , x ) ⁢ d x for t ∈ [ 0 , T ] , and applying (1.6), we obtain: ν ′ ⁢ ( t ) ≤ 2 ⁢ ( 1 + ln ⁡ ( c ) ) ⁢ ν ⁢ ( t ) for t ∈ [ 0 , T ] , which easily yields that

(1.9) ν ⁢ ( t ) ≤ e 2 ⁢ ( 1 + ln ⁡ ( c ) ) ⁢ t ⁢ ν ⁢ ( 0 ) for all ⁢ t ∈ [ 0 , T ] .

Since ν ⁢ ( 0 ) = 0 , from (1.9), we conclude that ν ⁢ ( t ) = 0 for t ∈ [ 0 , T ] , which is equivalent to u = w . ∎

For recent mathematical results, related to the problem above, we refer the reader to [5, 1, 10, 12, 7, 17, 14], while the authors are not aware of any published research work dealing with the numerical approximation of the solution to the heat equation with logarithmic nonlinearity. However, there are recent contributions to the numerical approximation of the solution to the logarithmic Schrödinger equation (see, e.g., [3, 2, 4, 13, 6, 19]), where the authors build up a numerical method for the underlying partial differential equation after changing it by substituting the logarithmic term g ⁢ ( z ) by g ε ⁢ ( z ) = z ⁢ ln ⁡ ( ε + | z | ) , where 𝜀 is a positive parameter close to zero. The advantage of this change is that the nonlinearity g ε of the new 𝜀-regularized partial differential equation is globally Lipschitz on ℂ with constant O ⁢ ( | ln ⁡ ( ε ) | ) , and thus existing techniques can be used to estimate the numerical approximation error which is added to the O ⁢ ( ε ) modelling error (see [2, Proposition 2.5]) caused by the change of the partial differential equation. However, this approach is not cloudless. Even though the regularization parameter 𝜀 acts as a discretization parameter along with the time step and the space-mesh width, the convergence analysis of the proposed numerical methods (see [3, 2, 4, 6]) arrives at error estimates suffering by constants that grow fast to infinity when 𝜀 tends to zero. In particular, we refer to [2, Theorem 3.1] for the presence of the term | ln ⁡ ( ε ) | 2 in the exponential constant that appears as an outcome of the application of the discrete Gronwall argument, when the error of a second-order linearly implicit finite difference method is estimated. We also refer to [3, Theorem 1], [4, Remark 4.7, Theorems 4.2 and 4.5] and [6, Theorem 2] for error estimates of the form O ⁢ ( | ln ⁡ ( ε ) | ⁢ τ 1 2 ) , O ⁢ ( τ ε ) or O ⁢ ( τ 2 ε 3 ) , when the convergence of linearly implicit splitting time-discrete methods with time step 𝜏 are analysed. Taking into account the O ⁢ ( ε ) modelling error, the latter results indicate that the order of convergence cannot be improved because higher order of convergence requires higher regularity for the 𝜀-dependent solution of the 𝜀-regularized partial differential equation that introduces a higher power of 1 ε in the error estimate. In particular, the higher possible rate of convergence with respect to 𝜏 is equal to 1 2 and can be obtained by choosing ε = τ 1 2 . A similar fact was expected in the results of [2], but it is hidden under the assumption that higher-order derivatives of the 𝜀-dependent solution are uniformly bounded by constants independent of 𝜀 (see [2, Theorem 3.1]). Thus, the convergence analysis leaves the impression that the time step and the space-mesh width are fighting against the influence of 𝜀 in the race of convergence. However, this situation is not confirmed by the presented numerical results and motivated our research on the numerical treatment of the logarithmic nonlinearity.

As a first step, we focused on the approximation of the solution to the logarithmic Schrödinger equation, over the two-dimensional rectangular domain 𝖣, by the Crank–Nicolson finite difference method [16]. Observing that g ε is an O ⁢ ( ε ) approximation of 𝑔 over ℂ, and using the Cazenave–Haraux property

| Im ⁡ [ ( g ε ⁢ ( z ) − g ε ⁢ ( w ) ) ⁢ ( z − w ¯ ) ] | ≤ | z − w | 2 for all ⁢ z , w ∈ C ⁢ and all ⁢ ε ≥ 0 ,

along with the 𝐵-stability property of the Crank–Nicolson method, we were able to develop a convergence analysis in the discrete L t ∞ ⁢ ( L x 2 ) -norm arriving at an almost second-order error estimate of the form

O ⁢ ( τ 2 ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 )

and of the form O ⁢ ( ε + τ 2 ⁢ | ln ⁡ ( ε ) | + h 1 2 + h 2 2 ) when 𝑔 is approximated by g ε in the numerical method (see [16, Theorem 4.1]). (Here, 𝜏 is the time step, and h 1 and h 2 are the mesh length of a uniform partition of the intervals [ a 1 , a 2 ] and [ b 1 , b 2 ] , respectively.) Thus, we achieved the Gronwall constant free of 𝜀 and avoided the presence of negative powers of 𝜀 in the error estimate. This is evidence that the obtained error estimates depend on the stability properties of the employed numerical method and on the structure of the convergence analysis.

In the paper at hand, we consider the semilinear heat equation with logarithmic nonlinearity (1.1) over a two-dimensional rectangular domain. Since 𝑔 is not differentiable or Lipschitz on ℝ, we are interested in the approximation of the solution 𝑢 by an implicit-explicit time-stepping method, in order to avoid the use of iterative methods solving numerically nonlinear systems of algebraic equations. Moving to this direction, after adopting a standard second-order finite difference method for space discretization, we employ for time stepping a linearized backward Euler method (see (2.6)–(2.7)), or a linearized BDF2 method (see (2.8)–(2.11)), to arrive at the (LBEFD) or (LBDF2FD) method, respectively. Both methods are implicit-explicit since they treat the linear part of the equation implicitly and its logarithmic nonlinearity g ⁢ ( u ) explicitly. The estimation of their consistency error was done first by approximating 𝑔 by g τ 2 and then applying the Taylor formula in the standard way, assuming that the solution 𝑢 is smooth enough on 𝖰. To build up a convergence analysis, first we construct a modified version of each numerical method (see Sections 5 and 6), by properly applying a parameter-dependent mollifier which is a smooth cut-off of the identity function (cf. [20]). Then we carry out an error analysis of the modified version of the proposed numerical methods, which is based on the strong stability property of each method and on a global one-sided Lipschitz condition for the composition of the parameter-dependent mollifier into g τ 2 (see Lemma 3.2) and arrives at almost optimal order error estimates. Finally, we are able to obtain the same error estimates for the proposed methods because, by construction, when the modified (LBEFD) or (LBDF2FD) approximations are close to the solution 𝑢, then they coincide with those of the (LBEFD) or (LBDF2FD) method, respectively. In particular, for the (LBEFD) method, without imposing mesh conditions, we provide an almost optimal order error estimate in the discrete L t ∞ ⁢ ( L x ∞ ) -norm of the form

max 0 ≤ n ≤ N | U n − u n | ∞ , H ≤ C LBEFD [ τ | ln ( τ ) | + h 1 2 + h 2 2 ] ,

and for the (LBDF2FD) method, after imposing a mild mesh condition (see (6.29)), we establish error estimates, in the discrete L t ∞ ⁢ ( L x 2 ) norm and in the discrete L t ∞ ⁢ ( H x 1 ) -norm, of the form

max 0 ≤ n ≤ N ∥ U n − u n ∥ 0 , H ≤ C LBDF2FD , 1 ⁢ [ τ 2 ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ] , max 0 ≤ n ≤ N | U n − u n | 1 , H ≤ C LBDF2FD , 2 ⁢ [ τ 2 ⁢ | ln ⁡ ( τ ) | 2 + ( h 1 2 + h 2 2 ) ⁢ | ln ⁡ ( τ ) | ] ,

where | ⋅ | ∞ , H is a discrete L ∞ ⁢ ( D ) norm, ∥ ⋅ ∥ 0 , H is a discrete L 2 ⁢ ( D ) norm, | ⋅ | 1 , H is a discrete H 1 ⁢ ( D ) norm, and C LBEFD , C LBDF2FD , 1 and C LBDF2FD , 2 are constants independent of 𝜏, h 1 and h 2 . We would like to note that similar error estimates can be obtained when the finite difference method is coupled with the backward Euler, the Crank–Nicolson method or the BDF2 method (see, e.g., [15]), and analogous convergence results can be derived when the (LBEFD) or the (LBDF2FD) method is applied to approximate the solution to the logarithmic Schrödinger equation.

We close this section by giving a synopsis of the paper. In Section 2, we set up notation, provide a series of helpful results and formulate the implicit-explicit finite difference methods we propose. In Section 3, we give the definition along with the basic properties of some parameter-dependent auxiliary functions. Section 4, includes the formulation and the estimation of the consistency error of our numerical methods, and Sections 5 and 6 are dedicated to their the convergence analysis. Finally, we present results from numerical experiments in Section 7.

2 The Implicit-Explicit Finite Difference Methods

2.1 Preliminaries

Let ℕ be the set of all positive integers. For given N ∈ N , we introduce a uniform partition of the time interval [ 0 , T ] with time step τ := T N and nodes t n := n ⁢ τ for n = 0 , … , N . Also, for given J 1 , J 2 ∈ N , we define a uniform partition of [ a 1 , a 2 ] with mesh width h 1 := a 2 − a 1 J 1 + 1 and nodes x 1 , i := a 1 + i ⁢ h 1 for i = 0 , … , J 1 + 1 , along with a uniform partition of [ b 1 , b 2 ] with mesh width h 2 := b 2 − b 1 J 2 + 1 and nodes x 2 , j := b 1 + j ⁢ h 2 for j = 0 , … , J 2 + 1 . To simplify the notation, we set I := { ( i , j ) : i = 0 , … , J 1 + 1 , j = 0 , … , J 2 + 1 } , I ∘ := { ( i , j ) : i = 1 , … , J 1 , j = 1 , … , J 2 } and ∂ I := I \ I ∘ . Then we introduce the discrete matrix space

X H := { ( V α ) α ∈ I ∈ R ( J 1 + 2 ) × ( J 2 + 2 ) : V α = 0 ⁢ for all ⁢ α ∈ ∂ I } ,

a discrete Laplacian operator Δ H : X H ↦ X H by

( Δ H ⁢ V ) ( i , j ) := V ( i − 1 , j ) − 2 ⁢ V ( i , j ) + V ( i + 1 , j ) h 1 2 + V ( i , j − 1 ) − 2 ⁢ V ( i , j ) + V ( i , j + 1 ) h 2 2 for all ⁢ ( i , j ) ∈ I ∘ ⁢ and all ⁢ V ∈ X H

and the operator I H : C ⁢ ( D ) ↦ X H by ( I H ⁢ [ z ] ) α := z ⁢ ( x 1 , α 1 , x 2 , α 2 ) for all α ∈ I ∘ and z ∈ C ⁢ ( D ) .

We define on X H a discrete L ∞ ⁢ ( D ) -norm | ⋅ | ∞ , H by | Z | ∞ , H := max α ∈ I ∘ ⁡ | Z α | for Z ∈ X H , a discrete L 2 ⁢ ( D ) -inner product ( ⋅ , ⋅ ) 0 , H by ( V , Z ) 0 , H := h 1 ⁢ h 2 ⁢ ∑ α ∈ I ∘ V α ⁢ Z α for V , Z ∈ X H , and a discrete H 1 ⁢ ( D ) -inner product ( ⋅ , ⋅ ) 1 , H by

( V , Z ) 1 , H := h 1 ⁢ h 2 ⁢ [ ∑ j = 1 J 2 ∑ i = 0 J 1 V ( i + 1 , j ) − V ( i , j ) h 1 ⁢ Z ( i + 1 , j ) − Z ( i , j ) h 1 + ∑ i = 1 J 1 ∑ j = 0 J 2 V ( i , j + 1 ) − V ( i , j ) h 2 ⁢ Z ( i , j + 1 ) − Z ( i , j ) h 2 ]

for V , Z ∈ X H . Also, we denote by ∥ ⋅ ∥ 0 , H and | ⋅ | 1 , H the corresponding norms, i.e. ∥ V ∥ 0 , H := [ ( V , V ) 0 , H ] 1 / 2 and | V | 1 , H := [ ( V , V ) 1 , H ] 1 / 2 for V ∈ X H .

Later, we will use the discrete integration-by-parts result

(2.1) ( Δ H ⁢ V , V ) 0 , H = − | V | 1 , H 2 for all ⁢ V ∈ X H ,

the discrete Poincaré–Friedrichs inequality

(2.2) ∥ V ∥ 0 , H ≤ 1 2 ⁢ min ⁡ { a 2 − a 1 , b 2 − b 1 } ⁢ | V | 1 , H for all ⁢ V ∈ X H

and the following inverse inequality.

Lemma 2.1

For V ∈ X H , it holds that

(2.3) | V | ∞ , H ≤ L h 1 + h 2 ⁢ | V | 1 , H

with

L := [ ( a 2 − a 1 ) ⁢ ( b 2 − b 1 ) min ⁡ { a 2 − a 1 , b 2 − b 1 } ] 1 / 2 = [ max ⁡ { a 2 − a 1 , b 2 − b 1 } ] 1 / 2 .

Proof

Let V ∈ X H and ( i 0 , j 0 ) ∈ I ∘ such that | V | ∞ , H = | V ( i 0 , j 0 ) | . Since V ( 0 , j 0 ) = V ( i 0 , 0 ) = 0 , using the Cauchy–Schwarz inequality, we obtain

(2.4) | V ( i 0 , j 0 ) | 2 ≤ ( a 2 − a 1 ) ⁢ h 1 ⁢ ∑ i = 0 J 1 | V ( i + 1 , j 0 ) − V ( i , j 0 ) h 1 | 2 ≤ a 2 − a 1 h 2 ⁢ h 1 ⁢ h 2 ⁢ ∑ j = 1 J 2 ∑ i = 0 J 1 | V ( i + 1 , j ) − V ( i , j ) h 1 | 2 ,

(2.5) | V ( i 0 , j 0 ) | 2 ≤ ( b 2 − b 1 ) ⁢ h 2 ⁢ ∑ j = 0 J 2 | V ( i 0 , j + 1 ) − V ( i 0 , j ) h 2 | 2 ≤ b 2 − b 1 h 1 ⁢ h 1 ⁢ h 2 ⁢ ∑ i = 1 J 1 ∑ j = 0 J 2 | V ( i , j + 1 ) − V ( i , j ) h 2 | 2 .

Combining (2.4) and (2.5), we conclude that

| V | ∞ , H 2 ≤ ( b 2 − b 1 ) ⁢ ( a 2 − a 1 ) ( a 2 − a 1 ) ⁢ h 1 + ( b 2 − b 1 ) ⁢ h 2 ⁢ | V | 1 , H 2 ,

which easily yields (2.3). ∎

Finally, we simplify the notation by setting u 1 / 2 := I H ⁢ [ u ⁢ ( τ 2 , ⋅ ) ] , u n := I H ⁢ [ u ⁢ ( t n , ⋅ ) ] for n = 0 , … , N , and by defining, for any W ∈ X H and g ∈ C ⁢ ( R ) , g ⁢ ( W ) ∈ X H by ( g ⁢ ( W ) ) α := g ⁢ ( W α ) for all α ∈ I ∘ .

2.2 Formulation of the Numerical Methods

For the approximation of the solution 𝑢 to problem (1.1)–(1.5), we formulate below two implicit-explicit finite difference methods requiring at every time step the solution of a linear, block-tridiagonal system of algebraic equations.

2.2.1 The (LBEFD) Method

The Linearized Backward Euler Finite Difference (LBEFD) method is a one-step method, and its algorithm is as follows.

Set
(2.6) U 0 := u 0 ∈ X H .
For n = 0 , … , N − 1 , find U n + 1 ∈ X H such that
(2.7) U n + 1 − U n = τ ⁢ Δ H ⁢ ( U n + 1 ) + τ ⁢ g ⁢ ( U n ) + τ ⁢ I H ⁢ [ f ⁢ ( t n + 1 , ⋅ ) ] .

2.2.2 The (LBDF2FD) Method

The Linearized BDF2 Finite Difference (LBDF2FD) method is a two-step method and has the following structure.

Set
(2.8) Υ 0 := u 0 ∈ X H .
Find Υ 1 / 2 ∈ X H such that
(2.9) Υ 1 / 2 − Υ 0 = τ 2 ⁢ Δ H ⁢ ( Υ 1 / 2 ) + τ 2 ⁢ g ⁢ ( Υ 0 ) + τ 2 ⁢ I H ⁢ [ f ⁢ ( τ 2 , ⋅ ) ] ,
and then find Υ 1 ∈ X H such that
(2.10) Υ 1 − Υ 0 = τ ⁢ Δ H ⁢ ( Υ 1 + Υ 0 2 ) + τ ⁢ g ⁢ ( Υ 1 / 2 ) + τ ⁢ I H ⁢ [ f ⁢ ( τ 2 , ⋅ ) ] .
For n = 0 , … , N − 2 , find Υ n + 2 ∈ X H such that
(2.11) 3 ⁢ Υ n + 2 − 4 ⁢ Υ n + 1 + Υ n = 2 ⁢ τ ⁢ Δ H ⁢ ( Υ n + 2 ) + 2 ⁢ τ ⁢ g ⁢ ( 2 ⁢ Υ n + 1 − Υ n ) + 2 ⁢ τ ⁢ I H ⁢ [ f ⁢ ( t n + 2 , ⋅ ) ] .

Remark 2.1

Existence and uniqueness of the (LBEFD) and (LBDF2FD) approximations follows with no conditions on 𝜏, h 1 and h 2 since, for any β > 0 , the linear operator G : X H ↦ X H , with G ⁢ ( V ) := V − β ⁢ τ ⁢ Δ H ⁢ V for all V ∈ X H , is invertible. The latter is an obvious outcome of the property ( G ⁢ ( V ) , V ) 0 , H = ∥ V ∥ 0 , H 2 + β ⁢ τ ⁢ | V | 1 , H 2 for all V ∈ X H , which is based on (2.1). Also, we note that the matrix of the corresponding linear system is symmetric and positive definite.

3 Auxiliary Functions

3.1 A 𝛿-Mollifier

For δ > 0 , let n δ ∈ C 1 ⁢ ( R ) (see, e.g., [20]) be an odd function defined by

(3.1) n δ ⁢ ( s ) := { s if ⁢ s ∈ [ 0 , δ ] , q δ ⁢ ( s ) if ⁢ s ∈ ( δ , 2 ⁢ δ ] , 2 ⁢ δ if ⁢ s > 2 ⁢ δ , for all ⁢ s ≥ 0 ,

where q δ ∈ P 3 ⁢ [ δ , 2 ⁢ δ ] is a polynomial defined by

(3.2) q δ ⁢ ( s ) := s + ( s − δ ) 2 ⁢ ( 2 ⁢ δ − s ) δ 2 > 0 for all ⁢ s ∈ [ δ , 2 ⁢ δ ] .

Some useful properties of the function n δ are exposed in the lemma below.

Lemma 3.1

For δ > 0 , it holds that

(3.3) n δ ′ ⁢ ( s ) ∈ [ 0 , 4 3 ] for all ⁢ s ∈ R

and

(3.4) max s ∈ R ⁡ | n δ ⁢ ( s ) | = 2 ⁢ δ .

Proof

Since n δ is an odd function, we conclude that n δ ′ is an even function, and thus it is sufficient to investigate the range of n δ ′ on [ 0 , + ∞ ) . According to (3.1), we have

(3.5) n δ ′ ⁢ ( s ) = 0 ⁢ for all ⁢ s ∈ [ 2 ⁢ δ , + ∞ ) and n δ ′ ⁢ ( s ) = 1 ⁢ for all ⁢ s ∈ [ 0 , δ ] .

Also, from (3.1) and (3.2), we obtain

n δ ′ ⁢ ( s ) = q δ ′ ⁢ ( s ) = δ 2 + ( s − δ ) ⁢ ( 5 ⁢ δ − 3 ⁢ s ) δ 2 = − 3 ⁢ s 2 + 8 ⁢ δ ⁢ s − 4 ⁢ δ 2 δ 2 for all ⁢ s ∈ [ δ , 2 ⁢ δ ] .

Observing that q δ ′ ⁢ ( δ ) = 1 , q δ ′ ⁢ ( 2 ⁢ δ ) = 0 and q δ ′′ ⁢ ( s ) ≥ 0 if and only if s ≤ 4 3 ⁢ δ , we conclude that q δ ′ is increasing on [ δ , 4 ⁢ δ 3 ] and decreasing on [ 4 ⁢ δ 3 , 2 ⁢ δ ] . Since n δ ′ ⁢ ( 4 ⁢ δ 3 ) = 4 3 , we easily arrive at

(3.6) n δ ′ ⁢ ( s ) ∈ [ 0 , 4 3 ] for all ⁢ s ∈ [ δ , 2 ⁢ δ ] .

Thus, (3.3) follows as a simple outcome of (3.5) and (3.6).

Finally, (3.3) yields that n δ is increasing on ℝ, and hence n δ ⁢ ( s ) ∈ [ 0 , 2 ⁢ δ ] for s ∈ [ 0 , + ∞ ) , from which (3.4) easily follows. ∎

Remark 3.1

Obviously, it holds that n δ ⁢ ( u ⁢ ( t , x ) ) = u ⁢ ( t , x ) for ( t , x ) ∈ Q , when δ ≥ max Q ⁡ | u | .

3.2 An 𝜀-Approximation of 𝑔

For ε > 0 , we define a function g ε : R → R by

g ε ⁢ ( s ) := s ⁢ ln ⁡ ( ε + | s | ) for all ⁢ s ∈ R .

It is well known (see, e.g., [16, Lemmas 2.1 and 2.2]) that

(3.7) sup s ∈ R | g ⁢ ( s ) − g ε ⁢ ( s ) | ≤ ε for all ⁢ ε > 0 ,

and

(3.8) | g ε ⁢ ( x ) − g ε ⁢ ( y ) | ≤ 2 ⁢ | ln ⁡ ( ε ) | ⁢ | x − y | for all ⁢ x , y ∈ [ − c , c ] ,

when c > e and ε ∈ ( 0 , 1 2 ⁢ c ) . Below, we show that the function g ε ∘ n δ (where ( g ε ∘ n δ ) ( s ) := g ε ( n δ ( s ) ) ) satisfies the following one-sided Lipschitz condition.

Lemma 3.2

Let ε ∈ ( 0 , 1 2 ⁢ e ) and δ > 0 . Then it holds that

(3.9) ( g ε ⁢ ( n δ ⁢ ( x ) ) − g ε ⁢ ( n δ ⁢ ( y ) ) ) ⁢ ( x − y ) ≤ 8 3 ⁢ ln ⁡ ( e + 2 ⁢ δ ) ⁢ | x − y | 2 for all ⁢ x , y ∈ R ,

(3.10) ( g ε ⁢ ( n δ ⁢ ( V ) ) − g ε ⁢ ( n δ ⁢ ( W ) ) , V − W ) 0 , H ≤ 8 3 ⁢ ln ⁡ ( e + 2 ⁢ δ ) ⁢ ∥ V − W ∥ 0 , H 2 for all ⁢ V , W ∈ X H .

Proof

Let x , y ∈ R with | n δ ⁢ ( y ) | ≤ | n δ ⁢ ( x ) | and S ⁢ ( x , y ) := ( g ε ⁢ ( n δ ⁢ ( x ) ) − g ε ⁢ ( n δ ⁢ ( y ) ) ) ⁢ ( x − y ) . The mean value theorem yields that there exist ξ ∈ [ ε + | n δ ⁢ ( y ) | , ε + | n δ ⁢ ( x ) | ] and z ∈ [ min ⁡ { x , y } , max ⁡ { x , y } ] such that

(3.11) ln ⁡ ( ε + | n δ ⁢ ( x ) | ) − ln ⁡ ( ε + | n δ ⁢ ( y ) | ) = 1 ξ ⁢ ( | n δ ⁢ ( x ) | − | n δ ⁢ ( y ) | )

and

(3.12) n δ ⁢ ( x ) − n δ ⁢ ( y ) = n δ ′ ⁢ ( z ) ⁢ ( x − y ) .

Now, using (3.12), (3.11), (3.3) and (3.4), we obtain

S ⁢ ( x , y ) = ( x − y ) ⁢ ( n δ ⁢ ( x ) − n δ ⁢ ( y ) ) ⁢ ln ⁡ ( ε + | n δ ⁢ ( x ) | ) + n δ ⁢ ( y ) ⁢ ( x − y ) ⁢ [ ln ⁡ ( ε + | n δ ⁢ ( x ) | ) − ln ⁡ ( ε + | n δ ⁢ ( y ) | ) ] = ( x − y ) 2 ⁢ n δ ′ ⁢ ( z ) ⁢ ln ⁡ ( ε + | n δ ⁢ ( x ) | ) + n δ ⁢ ( y ) ξ ⁢ ( x − y ) ⁢ ( | n δ ⁢ ( x ) | − | n δ ⁢ ( y ) | ) ≤ ( x − y ) 2 ⁢ n δ ′ ⁢ ( z ) ⁢ ln ⁡ ( e + 2 ⁢ δ ) + | n δ ⁢ ( y ) | ξ ⁢ | x − y | ⁢ | n δ ⁢ ( x ) − n δ ⁢ ( y ) | ≤ ( x − y ) 2 ⁢ n δ ′ ⁢ ( z ) ⁢ ln ⁡ ( e + 2 ⁢ δ ) + | n δ ⁢ ( y ) | ε + | n δ ⁢ ( y ) | ⁢ n δ ′ ⁢ ( z ) ⁢ | x − y | 2 ≤ 4 3 ⁢ | x − y | 2 ⁢ ( 1 + ln ⁡ ( e + 2 ⁢ δ ) ) .

Since (3.9) is symmetric with respect to 𝑥 and 𝑦, it holds also when | n δ ⁢ ( y ) | > | n δ ⁢ ( x ) | .

Finally, (3.10) follows as a simple consequence of (3.9). ∎

4 Consistency Errors

4.1 Time-Discretization Consistency Error

Let ( ρ n ) n = 0 N − 1 ⊂ X H be defined by

(4.1) u n + 1 − u n τ = I H ⁢ [ Δ ⁢ u ⁢ ( t n + 1 , ⋅ ) + g τ 2 ⁢ ( u ⁢ ( t n , ⋅ ) ) + f ⁢ ( t n + 1 , ⋅ ) ] + ρ n , n = 0 , … , N − 1 ,

and let ( r n ) n = 0 N ⊂ X H be given by

(4.2) u 1 / 2 − u 0 ( τ / 2 ) = I H ⁢ [ Δ ⁢ u ⁢ ( τ 2 , ⋅ ) + g ⁢ ( u 0 ⁢ ( ⋅ ) ) + f ⁢ ( τ 2 , ⋅ ) ] + r 0 ,

(4.3) u 1 − u 0 τ = I H ⁢ [ Δ ⁢ ( u ⁢ ( t 1 , ⋅ ) + u ⁢ ( t 0 , ⋅ ) 2 ) + g τ 2 ⁢ ( u ⁢ ( τ 2 , ⋅ ) ) + f ⁢ ( τ 2 , ⋅ ) ] + r 1 ,

(4.4) 3 ⁢ u n + 2 − 4 ⁢ u n + 1 + u n 2 ⁢ τ = I H ⁢ [ Δ ⁢ u ⁢ ( t n + 2 , ⋅ ) + g τ 2 ⁢ ( 2 ⁢ u ⁢ ( t n + 1 , ⋅ ) − u ⁢ ( t n , ⋅ ) ) ] + I H ⁢ [ f ⁢ ( t n + 2 , ⋅ ) ] + r n + 2 , n = 0 , … , N − 2 .

Combining (1.1), (4.1), (4.2), (4.3) and (4.4), we obtain

ρ n = [ u n + 1 − u n τ − I H ⁢ [ u t ⁢ ( t n + 1 , ⋅ ) ] ] − I H ⁢ [ g τ 2 ⁢ ( u ⁢ ( t n , ⋅ ) ) − g τ 2 ⁢ ( u ⁢ ( t n + 1 , ⋅ ) ) ] − I H ⁢ [ g τ 2 ⁢ ( u ⁢ ( t n + 1 , ⋅ ) ) − g ⁢ ( u ⁢ ( t n + 1 , ⋅ ) ) ] , n = 0 , … , N − 1 ,

r 0 = [ u 1 / 2 − u 0 ( τ / 2 ) − I H ⁢ [ u t ⁢ ( τ 2 , ⋅ ) ] ] − I H ⁢ [ g ⁢ ( u 0 ⁢ ( ⋅ ) ) − g τ 2 ⁢ ( u 0 ⁢ ( ⋅ ) ) ] − I H ⁢ [ g τ 2 ⁢ ( u 0 ⁢ ( ⋅ ) ) − g τ 2 ⁢ ( u ⁢ ( τ 2 , ⋅ ) ) ] − I H ⁢ [ g τ 2 ⁢ ( u ⁢ ( τ 2 , ⋅ ) ) − g ⁢ ( u ⁢ ( τ 2 , ⋅ ) ) ] ,

r 1 = [ u 1 − u 0 τ − I H ⁢ [ u t ⁢ ( τ 2 , ⋅ ) ] ] − I H ⁢ [ Δ ⁢ ( u ⁢ ( t 1 , ⋅ ) + u ⁢ ( t 0 , ⋅ ) 2 ) − Δ ⁢ u ⁢ ( τ 2 , ⋅ ) ] − I H ⁢ [ g τ 2 ⁢ ( u ⁢ ( τ 2 , ⋅ ) ) − g ⁢ ( u ⁢ ( τ 2 , ⋅ ) ) ] ,

r n + 2 = [ 3 ⁢ u n + 2 − 4 ⁢ u n + 1 + u n 2 ⁢ τ − I H ⁢ [ u t ⁢ ( t n + 2 , ⋅ ) ] ] − I H ⁢ [ g τ 2 ⁢ ( 2 ⁢ u ⁢ ( t n + 1 , ⋅ ) − u ⁢ ( t n , ⋅ ) ) − g τ 2 ⁢ ( u ⁢ ( t n + 2 , ⋅ ) ) ] − I H ⁢ [ g τ 2 ⁢ ( u ⁢ ( t n + 2 , ⋅ ) ) − g ⁢ ( u ⁢ ( t n + 2 , ⋅ ) ) ] , n = 0 , … , N − 2 .

Let c ⋆ := e + 3 ⁢ max Q ⁡ | u | and τ ∈ ( 0 , 1 2 ⁢ c ⋆ ) . Since τ < 1 , we have also τ 2 ∈ ( 0 , 1 2 ⁢ c ⋆ ) . Then, using the Taylor formula, the mean value theorem, (3.7) (with ε = τ 2 ) and (3.8) (with c = c ⋆ and ε = τ 2 ), we arrive at the following error bounds:

(4.5) | r 0 | ∞ , H + max 0 ≤ n ≤ N − 1 | ρ n | ∞ , H ≤ C TC1 [ τ max Q | ∂ t 2 u | + τ | ln ( τ ) | max Q | ∂ t u | + τ 2 ] ,

(4.6) | r 1 | ∞ , H ≤ C TC2 ⁢ [ τ 2 ⁢ max Q ⁡ | ∂ t 3 u | + τ 2 ⁢ max Q ⁡ | ∂ t 2 Δ ⁢ u | + τ 2 ] ,

(4.7) max 2 ≤ n ≤ N | r n | ∞ , H ≤ C TC3 [ τ 2 max Q | ∂ t 3 u | + τ 2 | ln ( τ ) | max Q | ∂ t 2 u | + τ 2 ] .

4.2 Space-Discretization Consistency Error

Let ( σ n ) n = 0 N − 1 ∈ X H be given by

(4.8) u n + 1 − u n τ = Δ H ⁢ ( u n + 1 ) + I H ⁢ [ g τ 2 ⁢ ( u ⁢ ( t n , ⋅ ) ) + f ⁢ ( t n + 1 , ⋅ ) ] + σ n , n = 0 , … , N − 1 ,

and let ( s m ) m = 0 N ⊂ X H be defined by

(4.9) u 1 / 2 − u 0 ( τ / 2 ) = Δ H ⁢ ( u 1 / 2 ) + I H ⁢ [ g ⁢ ( u 0 ⁢ ( ⋅ ) ) + f ⁢ ( τ 2 , ⋅ ) ] + s 0 ,

(4.10) u 1 − u 0 τ = Δ H ⁢ ( u 1 + u 0 2 ) + I H ⁢ [ g τ 2 ⁢ ( u ⁢ ( τ 2 , ⋅ ) ) + f ⁢ ( τ 2 , ⋅ ) ] + s 1

and

(4.11) 3 ⁢ u n + 2 − 4 ⁢ u n + 1 + u n 2 ⁢ τ = Δ H ⁢ ( u n + 2 ) + I H ⁢ [ g τ 2 ⁢ ( 2 ⁢ u ⁢ ( t n + 1 , ⋅ ) − u ⁢ ( t n , ⋅ ) ) ] + I H ⁢ [ f ⁢ ( t n + 2 , ⋅ ) ] + s n + 2 , n = 0 , … , N − 2 .

Then, subtracting (4.8) from (4.1), (4.9) from (4.2), (4.10) from (4.3) and (4.11) from (4.4), we obtain

σ n − ρ n = I H ⁢ [ Δ ⁢ u ⁢ ( t n + 1 , ⋅ ) ] − Δ H ⁢ ( u n + 1 ) , n = 0 , … , N − 1 , s 0 − r 0 = I H ⁢ [ Δ ⁢ u ⁢ ( τ 2 , ⋅ ) ] − Δ H ⁢ ( u 1 / 2 ) , s 1 − r 1 = I H ⁢ [ Δ ⁢ ( u ⁢ ( t 1 , ⋅ ) + u ⁢ ( t 0 , ⋅ ) 2 ) ] − Δ H ⁢ ( u 1 + u 0 2 ) , s n + 2 − r n + 2 = I H ⁢ [ Δ ⁢ u ⁢ ( t n + 2 , ⋅ ) ] − Δ H ⁢ ( u n + 2 ) , n = 0 , … , N − 2 .

After using the Taylor formula with respect to the space variables, we finally conclude that

(4.12) max { max 0 ≤ n ≤ N − 1 | σ n − ρ n | ∞ , H , max 0 ≤ n ≤ N | s n − r n | ∞ , H } ≤ C SC [ h 1 2 max Q | ∂ x 1 4 u | + h 2 2 max Q | ∂ x 2 4 u | ] .

5 Convergence of the (LBEFD) Method

For δ > 0 , the modified (LBEFD) approximations ( U δ n ) n = 0 N ⊂ X H of the solution 𝑢 are defined by the steps below.

Set
(5.1) U δ 0 := u 0 .
For n = 0 , … , N − 1 , find U δ n + 1 ∈ X H such that
(5.2) U δ n + 1 − U δ n = τ ⁢ Δ H ⁢ ( U δ n + 1 ) + τ ⁢ g ⁢ ( n δ ⁢ ( U δ n ) ) + τ ⁢ I H ⁢ [ f ⁢ ( t n + 1 , ⋅ ) ] .

Remark 5.1

Remark 2.1 yields that the existence and uniqueness of the modified (LBEFD) approximations. Also, letting δ ≥ max Q ⁡ | u | , it follows that | U 0 | ∞ , H ≤ δ , and thus, in view of Remark 3.1, we obtain g ⁢ ( n δ ⁢ ( U 0 ) ) = g ⁢ ( U 0 ) . Then, from (5.2) and (2.7) (with n = 0 ), we conclude that U δ 1 = U 1 .

Let us now discuss the convergence properties of the modified (LBEFD) approximations.

Theorem 5.1

Let δ ⋆ = e + 3 ⁢ max Q ⁡ | u | , τ ∈ ( 0 , 1 4 ⁢ δ ⋆ ) and ( U δ ⋆ m ) m = 0 N be the corresponding modified (LBEFD) approximations defined by (5.1)–(5.2). Then there exists a positive constant C δ ⋆ L , independent of 𝜏, h 1 and h 2 , such that

(5.3) max 0 ≤ m ≤ N ∥ u m − U δ ⋆ m ∥ 0 , H + max 0 ≤ m ≤ N | u m − U δ ⋆ m | ∞ , H ≤ C δ ⋆ L ( τ | ln ( τ ) | + h 1 2 + h 2 2 ) .

Proof

We simplify the notation, by setting e m := u m − U δ ⋆ m for m = 0 , … , N . Also, we use the symbol 𝐶 to denote a generic non-negative constant that is independent of 𝜏, h 1 , h 2 and δ ⋆ and may change value from one line to the other, and the symbol C δ ⋆ to denote a generic non-negative constant that depends on δ ⋆ but is independent of 𝜏, h 1 and h 2 and may change value from one line to the other.

Subtract (5.2) from (4.8) to get, in the light of Remark 3.1, the following error equations:

(5.4) e n + 1 − e n = τ ⁢ Δ H ⁢ ( e n + 1 ) + τ ⁢ [ g τ 2 ⁢ ( n δ ⋆ ⁢ ( u n ) ) − g ⁢ ( n δ ⋆ ⁢ ( U δ ⋆ n ) ) ] + τ ⁢ σ n , n = 0 , … , N − 1 .

Discrete L 2 -error estimate. Take the ( ⋅ , ⋅ ) 0 , H -inner product of both sides of (5.4) with e n + 1 and then use (2.1) to arrive at

(5.5) ∥ e n + 1 ∥ 0 , H 2 − ∥ e n ∥ 0 , H 2 + ∥ e n + 1 − e n ∥ 0 , H 2 + 2 ⁢ τ ⁢ | e n + 1 | 1 , H 2 ≤ L 1 n + L 2 n + L 3 n , n = 0 , … , N − 1 ,

where

L 1 n := 2 ⁢ τ ⁢ ( σ n , e n + 1 ) 0 , H , L 2 n := 2 ⁢ τ ⁢ ( g τ 2 ⁢ ( n δ ⋆ ⁢ ( u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( U δ ⋆ n ) ) , e n + 1 ) 0 , H , L 3 n := 2 ⁢ τ ⁢ ( g τ 2 ⁢ ( n δ ⋆ ⁢ ( U δ ⋆ n ) ) − g ⁢ ( n δ ⋆ ⁢ ( U δ ⋆ n ) ) , e n + 1 ) 0 , H .

Let n ∈ { 0 , … , N − 1 } . First, we use the Cauchy–Schwarz inequality, (4.5), (4.12), (2.2) and the arithmetic mean inequality to have

(5.6) L 1 n ≤ 2 ⁢ τ ⁢ ∥ σ n ∥ 0 , H ⁢ ∥ e n + 1 ∥ 0 , H ≤ 2 ⁢ τ ⁢ ( ∥ σ n − ρ n ∥ 0 , H + ∥ ρ n ∥ 0 , H ) ⁢ ∥ e n + 1 ∥ 0 , H ≤ C ⁢ τ ⁢ ( h 1 2 + h 2 2 + τ ⁢ | ln ⁡ ( τ ) | ) ⁢ | e n + 1 | 1 , H ≤ C ⁢ τ ⁢ ( τ ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ) 2 + τ 2 ⁢ | e n + 1 | 1 , H 2 .

Also, we use the Cauchy–Schwarz inequality, (3.10) (with δ = δ ⋆ and ε = τ 2 ), (3.4), (3.3), the arithmetic mean inequality and (3.8) (with c = 2 ⁢ δ ⋆ and ε = τ 2 ) to get

L 2 n = 2 ⁢ τ ⁢ ( g τ 2 ⁢ ( n δ ⋆ ⁢ ( u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( U δ ⋆ n ) ) , e n + 1 − e n ) 0 , H + 2 ⁢ τ ⁢ ( g τ 2 ⁢ ( n δ ⋆ ⁢ ( u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( U δ ⋆ n ) ) , e n ) 0 , H

≤ 2 ⁢ τ ⁢ ∥ g τ 2 ⁢ ( n δ ⋆ ⁢ ( u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( U δ ⋆ n ) ) ∥ 0 , H ⁢ ∥ e n + 1 − e n ∥ 0 , H + 16 3 ⁢ τ ⁢ ln ⁡ ( e + 2 ⁢ δ ⋆ ) ⁢ ∥ e n ∥ 0 , H 2

≤ τ 2 ⁢ ∥ g τ 2 ⁢ ( n δ ⋆ ⁢ ( u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( U δ ⋆ n ) ) ∥ 0 , H 2 + ∥ e n + 1 − e n ∥ 0 , H 2 + C δ ⋆ ⁢ τ ⁢ ∥ e n ∥ 0 , H 2

≤ 16 τ 2 | ln ( τ ) | 2 max R | n δ ⋆ ′ | 2 ∥ e n ∥ 0 , H 2 + ∥ e n + 1 − e n ∥ 0 , H 2 + C δ ⋆ τ ∥ e n ∥ 0 , H 2

≤ C ⁢ τ 2 ⁢ | ln ⁡ ( τ ) | 2 ⁢ ∥ e n ∥ 0 , H 2 + ∥ e n + 1 − e n ∥ 0 , H 2 + C δ ⋆ ⁢ τ ⁢ ∥ e n ∥ 0 , H 2

(5.7) ≤ C δ ⋆ ⁢ τ ⁢ ( 1 + τ ⁢ | ln ⁡ ( τ ) | 2 ) ⁢ ∥ e n ∥ 0 , H 2 + ∥ e n + 1 − e n ∥ 0 , H 2 .

Finally, we use the Cauchy–Schwarz inequality, (2.2), (3.7) (with ε = τ 2 ) and the arithmetic mean inequality to obtain

(5.8) L 3 n ≤ 2 ⁢ τ ⁢ ∥ g τ 2 ⁢ ( n δ ⋆ ⁢ ( U δ ⋆ n ) ) − g ⁢ ( n δ ⋆ ⁢ ( U δ ⋆ n ) ) ∥ 0 , H ⁢ ∥ e n + 1 ∥ 0 , H ≤ C ⁢ τ 3 ⁢ | e n + 1 | 1 , H ≤ C ⁢ τ 5 + τ 2 ⁢ | e n + 1 | 1 , H 2 .

Observing that 0 < τ ⁢ | ln ⁡ ( τ ) | 2 ≤ 4 e 2 and combining (5.5), (5.6), (5.7) and (5.8), we conclude that

∥ e n + 1 ∥ 0 , H 2 ≤ ( 1 + C δ ⋆ ⁢ τ ) ⁢ ∥ e n ∥ 0 , H 2 + C δ ⋆ ⁢ τ ⁢ ( τ ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ) 2 , n = 0 , … , N − 1 .

Finally, we apply a standard discrete Gronwall argument and use that e 0 = 0 , to arrive at

(5.9) max 0 ≤ m ≤ N ∥ e m ∥ 0 , H ≤ C δ ⋆ ( τ | ln ( τ ) | + h 1 2 + h 2 2 ) .

Discrete L ∞ -error estimate. Let n ∈ { 0 , … , N − 1 } and α = ( α 1 , α 2 ) ∈ I ∘ such that | e α n + 1 | = | e n + 1 | ∞ , H . Multiplying both sides of (5.4) with e α n + 1 , we obtain

( 1 + 2 ⁢ τ h 1 2 + 2 ⁢ τ h 2 2 ) ⁢ ( e α n + 1 ) 2 = e α n ⁢ e α n + 1 + τ h 1 2 ⁢ [ e ( α 1 − 1 , α 2 ) n + 1 + e ( α 1 + 1 , α 2 ) n + 1 ] ⁢ e α n + 1 + τ h 2 2 ⁢ [ e ( α 1 , α 2 − 1 ) n + 1 + e ( α 1 , α 2 + 1 ) n + 1 ] ⁢ e α n + 1 + τ ⁢ [ g τ 2 ⁢ ( n δ ⋆ ⁢ ( u α n ) ) − g ⁢ ( n δ ⋆ ⁢ ( ( U δ ⋆ n ) α ) ) ] ⁢ e α n + 1 + τ ⁢ [ ( σ α n − ρ α n ) + ρ α n ] ⁢ e α n + 1 ,

which easily yields that

(5.10) | e α n + 1 | 2 + | e α n + 1 − e α n | 2 ≤ | e α n | 2 + Λ α 1 , n + Λ α 2 , n + Λ α 3 , n + Λ α 4 , n ,

where

Λ α 1 , n := 2 ⁢ τ ⁢ ( ( σ α n − ρ α n ) + ρ α n ) ⁢ e α n + 1 , Λ α 2 , n := 2 ⁢ τ ⁢ [ g τ 2 ⁢ ( n δ ⋆ ⁢ ( u α n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( ( U δ ⋆ n ) α ) ) ] ⁢ ( e α n + 1 − e α n ) , Λ α 3 , n := 2 ⁢ τ ⁢ [ g τ 2 ⁢ ( n δ ⋆ ⁢ ( u α n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( ( U δ ⋆ n ) α ) ) ] ⁢ e α n , Λ α 4 , n := 2 ⁢ τ ⁢ [ g τ 2 ⁢ ( n δ ⋆ ⁢ ( ( U δ ⋆ n ) α ) ) − g ⁢ ( n δ ⋆ ⁢ ( ( U δ ⋆ n ) α ) ) ] ⁢ e α n + 1 .

Using (4.5), (4.12), (3.7) (with ε = τ 2 ) and the arithmetic mean inequality, we have

(5.11) Λ α 1 , n ≤ C ⁢ τ ⁢ ( τ ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ) ⁢ | e n + 1 | ∞ , H ≤ C ⁢ τ ⁢ ( τ ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ) 2 + τ 4 ⁢ | e n + 1 | ∞ , H 2 ,

(5.12) Λ α 4 , n ≤ C ⁢ τ 3 ⁢ | e n + 1 | ∞ , H ≤ C ⁢ τ 5 + τ 4 ⁢ | e n + 1 | ∞ , H 2 .

Applying (3.9) (with δ = δ ⋆ and ε = τ 2 ) along with (3.3), (3.4), we conclude that

(5.13) Λ α 3 , n ≤ C ⁢ τ ⁢ ln ⁡ ( e + 2 ⁢ δ ⋆ ) ⁢ ( e α n ) 2 ≤ C δ ⋆ ⁢ τ ⁢ | e n | ∞ , H 2 .

Also, (3.3), (3.8) (with c = 2 ⁢ δ ⋆ and ε = τ 2 ) and the arithmetic mean inequality yield

(5.14) Λ α 2 , n ≤ 2 ⁢ τ ⁢ | g τ 2 ⁢ ( n δ ⋆ ⁢ ( u α n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( ( U δ ⋆ n ) α ) ) | ⁢ | e α n + 1 − e α n | ≤ 8 ⁢ τ ⁢ | ln ⁡ ( τ ) | ⁢ max R ⁡ | n δ ⋆ ′ | ⁢ | e α n | ⁢ | e α n + 1 − e α n | ≤ C ⁢ τ ⁢ | ln ⁡ ( τ ) | ⁢ | e α n | ⁢ | e α n + 1 − e α n | ≤ C δ ⋆ ⁢ τ 2 ⁢ | ln ⁡ ( τ ) | 2 ⁢ | e α n | 2 + | e α n + 1 − e α n | 2 ≤ C δ ⋆ ⁢ τ ⁢ ( τ ⁢ | ln ⁡ ( τ ) | 2 ) ⁢ | e n | ∞ , H 2 + | e α n + 1 − e α n | 2 .

Since 0 < τ ⁢ | ln ⁡ ( τ ) | 2 ≤ 4 e 2 , using (5.10), (5.11), (5.12), (5.13) and (5.14), we arrive at

( 1 − τ 2 ) ⁢ | e n + 1 | ∞ , H 2 ≤ ( 1 + C δ ⋆ ⁢ τ ) ⁢ | e n | ∞ , H 2 + C δ ⋆ ⁢ τ ⁢ ( τ ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ) 2

for n = 0 , … , N − 1 . Since τ < 1 and e 0 = 0 , a standard Gronwall argument yields

(5.15) max 0 ≤ m ≤ N | e m | ∞ , H ≤ C δ ⋆ ( τ | ln ( τ ) | + h 1 2 + h 2 2 ) .

Thus, (5.3) easily follows from (5.9) and (5.15). ∎

Next, we present a convergence result for the (LBEFD) method.

Theorem 5.2

Let δ ⋆ = e + 3 ⁢ max Q ⁡ | u | , τ ∈ ( 0 , 1 4 ⁢ δ ⋆ ) , let C δ ⋆ L be the positive constant specified in Theorem 5.1, let ( U δ ⋆ m ) m = 0 N be the modified (LBEFD) approximations defined by (5.1)–(5.2), and let ( U m ) m = 0 N be the (LBEFD) approximations defined by (2.6)–(2.7). If

(5.16) C δ ⋆ L ⁢ ( τ ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ) ≤ δ ⋆ 2 ,

then U m = U δ ⋆ m for m = 0 , … , N , and

(5.17) max 0 ≤ m ≤ N ∥ u m − U m ∥ 0 , H + max 0 ≤ m ≤ N | u m − U m | ∞ , H ≤ C δ ⋆ L ( τ | ln ( τ ) | + h 1 2 + h 2 2 ) .

Proof

Using that δ ⋆ > 3 ⁢ max Q ⁡ | u | , along with (5.3) and (5.16), for n = 1 , … , N − 1 , we obtain

| U δ ⋆ n | ∞ , H ≤ | u n − U δ ⋆ n | ∞ , H + | u n | ∞ , H < C δ ⋆ L ⁢ ( τ ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ) + δ ⋆ 3 ≤ δ ⋆ 2 + δ ⋆ 3 < δ ⋆ ,

which, in the light of (3.1), yields n δ ⋆ ⁢ ( U δ ⋆ n ) = U δ ⋆ n for n = 1 , … , N − 1 . Combining the latter relation with (5.1)–(5.2) and (2.6)–(2.7), we conclude that U δ ⋆ m = U m for m = 0 , … , N , and the error estimate (5.17) follows from (5.3). ∎

6 Convergence of the (LBDF2FD) Method

For δ > 0 , the modified (LBDF2FD) approximations ( Υ δ n ) n = 0 N ⊂ X H of the solution 𝑢 are defined as follows:

Set
(6.1) Υ δ 0 := u 0 , Υ δ 1 / 2 := Υ 1 / 2 ,
and find Υ δ 1 ∈ X H by
(6.2) Υ δ 1 − Υ δ 0 = τ ⁢ Δ H ⁢ ( Υ δ 1 + Υ δ 0 2 ) + τ ⁢ g ⁢ ( n δ ⁢ ( Υ δ 1 / 2 ) ) + τ ⁢ I H ⁢ [ f ⁢ ( τ 2 , ⋅ ) ] .
For n = 0 , … , N − 2 , find Υ δ n + 2 ∈ X H such that
(6.3) 3 ⁢ Υ δ n + 2 − 4 ⁢ Υ δ n + 1 + Υ δ n = 2 ⁢ τ ⁢ Δ H ⁢ ( Υ δ n + 2 ) + 2 ⁢ τ ⁢ g ⁢ ( n δ ⁢ ( 2 ⁢ Υ δ n + 1 − Υ δ n ) ) + 2 ⁢ τ ⁢ I H ⁢ [ f ⁢ ( t n + 2 , ⋅ ) ] .

Remark 6.1

The existence and uniqueness of the modified (LBDF2FD) approximations follows from Remark 2.1.

Let us investigate below the convergence of the modified (LBDF2FD) approximations.

Lemma 6.1

Let δ ⋆ = 2 ⁢ ( e + 3 ⁢ max Q ⁡ | u | ) , τ ∈ ( 0 , 1 4 ⁢ δ ⋆ ) , and let Υ 1 / 2 ∈ X H be specified by (2.9) and Υ δ ⋆ 1 ∈ X H by (6.2). Then there exist constants C B , 1 > 0 , C B , 2 > 0 and C B , 3 > 0 , independent of 𝜏, h 1 and h 2 , such that

(6.4) | u 1 / 2 − Υ 1 / 2 | ∞ , H ≤ C B , 1 ⁢ ( τ 2 ⁢ | ln ⁡ ( τ ) | + τ ⁢ h 1 2 + τ ⁢ h 2 2 ) ,

(6.5) ∥ u 1 − Υ δ ⋆ 1 ∥ 0 , H ≤ C B , 2 ⁢ ( τ 3 ⁢ | ln ⁡ ( τ ) | 2 + τ ⁢ h 1 2 + τ ⁢ h 2 2 ) ,

(6.6) | u 1 − Υ δ ⋆ 1 | 1 , H ≤ C B , 3 ⁢ τ 1 2 ⁢ ( τ 2 ⁢ | ln ⁡ ( τ ) | 2 + h 1 2 + h 2 2 ) .

Proof

Set e 1 / 2 := u 1 / 2 − Υ 1 / 2 and e 1 := u 1 − Υ δ ⋆ 1 . Combining (6.1), (2.9), (4.9), (6.2) and (4.10), we get the following error equations:

(6.7) e 1 / 2 = τ 2 ⁢ Δ H ⁢ ( e 1 / 2 ) + τ 2 ⁢ [ ( s 0 − r 0 ) + r 0 ] ,

(6.8) e 1 = τ 2 ⁢ Δ H ⁢ ( e 1 ) + τ ⁢ [ g τ 2 ⁢ ( u 1 / 2 ) − g ⁢ ( n δ ⋆ ⁢ ( Υ δ ⋆ 1 / 2 ) ) ] + τ ⁢ [ ( s 1 − r 1 ) + r 1 ] .

Proceeding as in the proof of Theorem 5.1 and using (4.5) and (4.12), we arrive at

(6.9) | e 1 / 2 | ∞ , H ≤ C ⁢ ( τ 2 ⁢ | ln ⁡ ( τ ) | + τ ⁢ h 1 2 + τ ⁢ h 2 2 ) .

Also, taking the ( ⋅ , ⋅ ) 0 , H -inner product of (6.8) with e 1 , and then applying (2.1), the Cauchy–Schwarz inequality and (3.1), we obtain

(6.10) ∥ e 1 ∥ 0 , H 2 + τ 2 ⁢ | e 1 | 1 , H 2 ≤ K 1 + K 2 + K 3 ,

where

K 1 := τ ⁢ ∥ g τ 2 ⁢ ( n δ ⋆ ⁢ ( u 1 / 2 ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( Υ 1 / 2 ) ) ∥ 0 , H ⁢ ∥ e 1 ∥ 0 , H , K 2 := τ ⁢ ∥ g τ 2 ⁢ ( n δ ⋆ ⁢ ( Υ 1 / 2 ) ) − g ⁢ ( n δ ⋆ ⁢ ( Υ 1 / 2 ) ) ∥ 0 , H ⁢ ∥ e 1 ∥ 0 , H , K 3 := τ ⁢ ( ∥ s 1 − r 1 ∥ 0 , H + ∥ r 1 ∥ 0 , H ) ⁢ ∥ e 1 ∥ 0 , H .

Using (4.12), (4.6), (3.7) (with ε = τ 2 ), (3.4), (3.3), (3.8) (with c = 2 ⁢ δ ⋆ and ε = τ 2 ) and (6.9), we get

(6.11) K 2 + K 3 ≤ C ⁢ [ τ 3 + τ ⁢ ( h 1 2 + h 2 2 ) ] ⁢ ∥ e 1 ∥ 0 , H

and

(6.12) K 1 ≤ C ⁢ τ ⁢ | ln ⁡ ( τ ) | ⁢ max R ⁡ | n δ ⋆ ′ | ⁢ ∥ e 1 / 2 ∥ 0 , H ⁢ ∥ e 1 ∥ 0 , H ≤ C ⁢ τ ⁢ | ln ⁡ ( τ ) | ⁢ | e 1 / 2 | ∞ , H ⁢ ∥ e 1 ∥ 0 , H ≤ C ⁢ [ τ 3 ⁢ | ln ⁡ ( τ ) | 2 + τ 2 ⁢ | ln ⁡ ( τ ) | ⁢ ( h 1 2 + h 2 2 ) ] ⁢ ∥ e 1 ∥ 0 , H ≤ C ⁢ [ τ 3 ⁢ | ln ⁡ ( τ ) | 2 + τ ⁢ ( h 1 2 + h 2 2 ) ] ⁢ ∥ e 1 ∥ 0 , H .

Now, from (6.10), (6.11) and (6.12), we arrive at

(6.13) ∥ e 1 ∥ 0 , H ≤ C ⁢ [ τ 3 ⁢ | ln ⁡ ( τ ) | 2 + τ ⁢ ( h 1 2 + h 2 2 ) ] ,

(6.14) | e 1 | 1 , H ≤ C ⁢ [ τ 5 2 ⁢ | ln ⁡ ( τ ) | 2 + τ 1 2 ⁢ ( h 1 2 + h 2 2 ) ] .

Thus, (6.4), (6.5) and (6.6) follow, respectively, from (6.9), (6.13) and (6.14). ∎

Theorem 6.1

Let δ ⋆ = 2 ⁢ ( e + 3 ⁢ max Q ⁡ | u | ) , τ ∈ ( 0 , 1 4 ⁢ δ ⋆ ) , and let ( Υ δ ⋆ m ) m = 0 N be the modified (LBDF2FD) approximations defined by (6.1)–(6.3). Then there exist constants C δ ⋆ F , 1 > 0 and C δ ⋆ F , 2 > 0 , independent of 𝜏, h 1 and h 2 , such that

(6.15) max 0 ≤ m ≤ N ∥ u m − Υ δ ⋆ m ∥ 0 , H ≤ C δ ⋆ F , 1 [ τ 2 | ln ( τ ) | + h 1 2 + h 2 2 ] ,

(6.16) max 0 ≤ m ≤ N | u m − Υ δ ⋆ m | 1 , H ≤ C δ ⋆ F , 2 [ τ 2 | ln ( τ ) | 2 + ( h 1 2 + h 2 2 ) | ln ( τ ) | ] .

Proof

We simplify the notation by setting e m := u m − Υ δ ⋆ m for m = 0 , … , N . We will use the symbol 𝐶 to denote a generic non-negative constant that is independent of 𝜏, h 1 , h 2 and δ ⋆ , and may change value from one line to the other, and the symbol C δ ⋆ to denote a generic non-negative constant that depends on δ ⋆ but is independent of 𝜏, h 1 and h 2 , and may change value from one line to the other. Also, we recall the following well-known algebraic identity:

(6.17) 2 ⁢ ( 3 ⁢ a − 4 ⁢ b + c ) ⁢ a = a 2 + | 2 ⁢ a − b | 2 − b 2 − | 2 ⁢ b − c | 2 + | a − 2 ⁢ b + c | 2 for all ⁢ a , b , c ∈ R ,

which is related to the 𝐺-stability property of the BDF2 method (see, e.g., [18, 8]).

Combining (6.3) with (4.11), we arrive at the corresponding error equations

(6.18) 3 ⁢ e n + 2 − 4 ⁢ e n + 1 + e n = 2 ⁢ τ ⁢ Δ H ⁢ ( e n + 2 ) + 2 ⁢ τ ⁢ s n + 2 + 2 ⁢ τ ⁢ [ g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ u n + 1 − u n ) ) − g ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) ]

for n = 0 , … , N − 2 .

Discrete L 2 -error estimate. Take the ( ⋅ , ⋅ ) 0 , H -inner product of both sides of (6.18) with 2 ⁢ e n + 2 , and then use (2.1) and (6.17) to arrive at

(6.19) ∥ e n + 2 ∥ 0 , H 2 + ∥ 2 ⁢ e n + 2 − e n + 1 ∥ 0 , H 2 = − ∥ e n + 2 − 2 ⁢ e n + 1 + e n ∥ 0 , H 2 − 4 ⁢ τ ⁢ | e n + 2 | 1 , H 2 + ∥ e n + 1 ∥ 0 , H 2 + ∥ 2 ⁢ e n + 1 − e n ∥ 0 , H 2 + Λ 1 n + Λ 2 n + Λ 3 n , n = 0 , … , N − 2 ,

where

Λ 1 n := 4 ⁢ τ ⁢ ( s n + 2 , e n + 2 ) 0 , H , Λ 2 n := 4 ⁢ τ ⁢ ( g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ u n + 1 − u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) , e n + 2 ) 0 , H , Λ 3 n := 4 ⁢ τ ⁢ ( g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) − g ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) , e n + 2 ) 0 , H .

Let n ∈ { 0 , … , N − 2 } . Using the Cauchy–Schwarz inequality, (4.7), (4.12), (2.2) and the arithmetic mean inequality, it follows that

(6.20) Λ 1 n ≤ 4 ⁢ τ ⁢ ∥ s n + 2 ∥ 0 , H ⁢ ∥ e n + 2 ∥ 0 , H ≤ 4 ⁢ τ ⁢ ( ∥ s n + 2 − r n + 2 ∥ 0 , H + ∥ r n + 2 ∥ 0 , H ) ⁢ ∥ e n + 2 ∥ 0 , H ≤ C ⁢ τ ⁢ ( h 1 2 + h 2 2 + τ 2 ⁢ | ln ⁡ ( τ ) | ) ⁢ | e n + 2 | 1 , H ≤ C ⁢ τ ⁢ ( τ 2 ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ) 2 + τ 2 ⁢ | e n + 2 | 1 , H 2 .

Applying the Cauchy–Schwarz inequality and combining (3.10) (with δ = δ ⋆ and ε = τ 2 ), (3.4), (3.3), the arithmetic mean inequality and (3.8) (with c = 2 ⁢ δ ⋆ and ε = τ 2 ), we obtain

(6.21) Λ 2 n = 4 ⁢ τ ⁢ ( g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ u n + 1 − u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) , e n + 2 − 2 ⁢ e n + 1 + e n ) 0 , H + 4 ⁢ τ ⁢ ( g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ u n + 1 − u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) , 2 ⁢ e n + 1 − e n ) 0 , H ≤ 4 ⁢ τ ⁢ ∥ g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ u n + 1 − u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) ∥ 0 , H ⁢ ∥ e n + 2 − 2 ⁢ e n + 1 + e n ∥ 0 , H + C ⁢ τ ⁢ ln ⁡ ( e + 2 ⁢ δ ⋆ ) ⁢ ∥ 2 ⁢ e n + 1 − e n ∥ 0 , H 2 ≤ 4 ⁢ τ 2 ⁢ ∥ g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ u n + 1 − u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) ∥ 0 , H 2 + ∥ e n + 2 − 2 ⁢ e n + 1 + e n ∥ 0 , H 2 + C δ ⋆ ⁢ τ ⁢ ∥ 2 ⁢ e n + 1 − e n ∥ 0 , H 2 ≤ C τ 2 | ln ( τ ) | 2 max R | n δ ⋆ ′ | 2 ∥ 2 e n + 1 − e n ∥ 0 , H 2 + ∥ e n + 2 − 2 e n + 1 + e n ∥ 0 , H 2 + C δ ⋆ τ ∥ 2 e n + 1 − e n ∥ 0 , H 2 ≤ C ⁢ τ 2 ⁢ | ln ⁡ ( τ ) | 2 ⁢ ∥ 2 ⁢ e n + 1 − e n ∥ 0 , H 2 + ∥ e n + 2 − 2 ⁢ e n + 1 + e n ∥ 0 , H 2 + C δ ⋆ ⁢ τ ⁢ ∥ 2 ⁢ e n + 1 − e n ∥ 0 , H 2 ≤ C δ ⋆ ⁢ τ ⁢ ( 1 + τ ⁢ | ln ⁡ ( τ ) | 2 ) ⁢ ∥ 2 ⁢ e n + 1 − e n ∥ 0 , H 2 + ∥ e n + 2 − 2 ⁢ e n + 1 + e n ∥ 0 , H 2 .

Applying again the Cauchy–Schwarz inequality, along with (2.2), (3.7) (with ε = τ 2 ) and the arithmetic mean inequality, we get

(6.22) Λ 3 n ≤ 4 ⁢ τ ⁢ ∥ g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) − g ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) ∥ 0 , H ⁢ ∥ e n + 2 ∥ 0 , H ≤ C ⁢ τ 3 ⁢ | e n + 2 | 1 , H ≤ C ⁢ τ 5 + τ 2 ⁢ | e n + 2 | 1 , H 2 .

Since τ ⁢ | ln ⁡ ( τ ) | 2 ≤ 4 e 2 , an obvious outcome of (6.19), (6.20), (6.21) and (6.22) is the following inequality:

∥ e n + 2 ∥ 0 , H 2 + ∥ 2 ⁢ e n + 2 − e n + 1 ∥ 0 , H 2 ≤ ( 1 + C δ ⋆ ⁢ τ ) ⁢ ( ∥ e n + 1 ∥ 0 , H 2 + ∥ 2 ⁢ e n + 1 − e n ∥ 0 , H 2 ) + C δ ⋆ ⁢ τ ⁢ ( τ 2 ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ) 2 , n = 0 , … , N − 2 .

Then, applying a standard discrete Gronwall argument and using that e 0 = 0 , we get

max 0 ≤ m ≤ N ∥ e m ∥ 0 , H ≤ C δ ⋆ [ ∥ e 1 ∥ 0 , H + τ 2 | ln ( τ ) | + h 1 2 + h 2 2 ]

which, along with (6.5), yields (6.15).

Discrete H 1 -error estimate. Take the ( ⋅ , ⋅ ) 0 , H -inner product of both sides of (6.18) with − 2 ⁢ Δ H ⁢ e n + 2 , and then use (2.1) and (6.17) to arrive at

(6.23) | e n + 2 | 1 , H 2 + | 2 ⁢ e n + 2 − e n + 1 | 1 , H 2 = | e n + 1 | 1 , H 2 + | 2 ⁢ e n + 1 − e n | 1 , H 2 − | e n + 2 − 2 ⁢ e n + 1 + e n | 1 , H 2 − 4 ⁢ τ ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H 2 + Z 1 n + Z 2 n + Z 3 n

for n = 0 , … , N − 2 , where

Z 1 n := − 4 ⁢ τ ⁢ ( s n + 2 , Δ H ⁢ e n + 2 ) 0 , H , Z 2 n := − 4 ⁢ τ ⁢ ( g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ u n + 1 − u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) , Δ H ⁢ e n + 2 ) 0 , H , Z 3 n := − 4 ⁢ τ ⁢ ( g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) − g ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) , Δ H ⁢ e n + 2 ) 0 , H .

Let n ∈ { 0 , … , N − 2 } . Using the Cauchy–Schwarz inequality, (4.7), (4.12) and the arithmetic mean inequality, it follows that

(6.24) Z 1 n ≤ 4 ⁢ τ ⁢ ∥ s n + 2 ∥ 0 , H ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H ≤ 4 ⁢ τ ⁢ ( ∥ s n + 2 − r n + 2 ∥ 0 , H + ∥ r n + 2 ∥ 0 , H ) ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H ≤ C ⁢ τ ⁢ ( h 1 2 + h 2 2 + τ 2 ⁢ | ln ⁡ ( τ ) | ) ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H ≤ C ⁢ τ ⁢ ( τ 2 ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ) 2 + τ 2 ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H 2 .

Using, again, (3.3), (3.4), (3.8) (with c = 2 ⁢ δ ⋆ and ε = τ 2 ), (3.7) (with ε = τ 2 ) and the arithmetic mean inequality, we obtain

(6.25) Z 2 n ≤ 4 ⁢ τ ⁢ ∥ g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ u n + 1 − u n ) ) − g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) ∥ 0 , H ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H ≤ C ⁢ τ ⁢ | ln ⁡ ( τ ) | ⁢ max R ⁡ | n δ ⋆ ′ | ⁢ ∥ 2 ⁢ e n + 1 − e n ∥ 0 , H ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H ≤ C ⁢ τ ⁢ | ln ⁡ ( τ ) | ⁢ ∥ 2 ⁢ e n + 1 − e n ∥ 0 , H ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H ≤ C ⁢ τ ⁢ | ln ⁡ ( τ ) | 2 ⁢ ( ∥ e n + 1 ∥ 0 , H 2 + ∥ e n ∥ 0 , H 2 ) + τ 2 ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H 2 ,

(6.26) Z 3 n ≤ 4 ⁢ τ ⁢ ∥ g τ 2 ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) − g ⁢ ( n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ n + 1 − Υ δ ⋆ n ) ) ∥ 0 , H ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H ≤ C ⁢ τ 3 ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H ≤ C ⁢ τ 5 + τ 2 ⁢ ∥ Δ H ⁢ e n + 2 ∥ 0 , H 2 .

Using (6.23), (6.24), (6.25), (6.26) and (6.15), we arrive at

| e n + 2 | 1 , H 2 + | 2 ⁢ e n + 2 − e n + 1 | 1 , H 2 = | e n + 1 | 1 , H 2 + | 2 ⁢ e n + 1 − e n | 1 , H 2 + C ⁢ τ ⁢ | ln ⁡ ( τ ) | 2 ⁢ [ τ 2 ⁢ | ln ⁡ ( τ ) | + h 1 2 + h 2 2 ] 2 , n = 0 , … , N − 2 .

Since e 0 = 0 , after using a standard Gronwall argument, we obtain

(6.27) max 0 ≤ m ≤ N | e m | 1 , H ≤ C δ ⋆ [ | e 1 | 1 , H + τ 2 | ln ( τ ) | 2 + ( h 1 2 + h 2 2 ) | ln ( τ ) | ] .

Thus, (6.16) easily follows from (6.27) and (6.6). ∎

Theorem 6.2

Let δ ⋆ = 2 ⁢ ( e + 3 ⁢ max Q ⁡ | u | ) , τ ∈ ( 0 , 1 4 ⁢ δ ⋆ ) , let C B , 1 be the constant introduced in Lemma 6.1, C δ ⋆ F , 1 , C δ ⋆ F , 2 the positive constants specified in Theorem 6.1, 𝖫 the constant specified in Lemma 2.1, and let ( Υ m ) m = 0 N be the (LBDF2FD) approximations defined by (2.8)–(2.11), and ( Υ δ ⋆ m ) m = 0 N the modified (LBDF2FD) approximations defined by (5.1)–(5.2). If

(6.28) C B , 1 ⁢ ( τ 2 ⁢ | ln ⁡ ( τ ) | + τ ⁢ h 1 2 + τ ⁢ h 2 2 ) ≤ δ ⋆ 2 ,

(6.29) 3 ⁢ C δ ⋆ F , 2 ⁢ L ⁢ ( h 1 + h 2 ) − 1 2 ⁢ [ τ 2 ⁢ | ln ⁡ ( τ ) | 2 + ( h 1 2 + h 2 2 ) ⁢ | ln ⁡ ( τ ) | ] ≤ δ ⋆ 6 ,

then Υ m = Υ δ ⋆ m for m = 0 , … , N , and

(6.30) max 0 ≤ m ≤ N ∥ u m − Υ m ∥ 0 , H ≤ C δ ⋆ F , 1 ( τ 2 | ln ( τ ) | + h 1 2 + h 2 2 ) ,

(6.31) max 0 ≤ m ≤ N | u m − Υ m | 1 , H ≤ C δ ⋆ F , 2 ( τ 2 | ln ( τ ) | 2 + ( h 1 2 + h 2 2 ) | ln ( τ ) | ) .

Proof

The assumption δ ⋆ > 6 ⁢ max Q ⁡ | u | , along with (6.4), (6.28), (6.16), (2.3) and (6.29), yields

| Υ 1 / 2 | ∞ , H ≤ | u 1 / 2 − Υ 1 / 2 | ∞ , H + | u 1 / 2 | ∞ , H < C B , 1 ⁢ [ τ 2 ⁢ | ln ⁡ ( τ ) | + τ ⁢ ( h 1 2 + h 2 2 ) ] + δ ⋆ 6 ≤ δ ⋆ 2 + δ ⋆ 6 < δ ⋆ ,

and

| 2 ⁢ Υ δ ⋆ m + 1 − Υ δ ⋆ m | ∞ , H ≤ 2 ⁢ | u m + 1 − Υ δ ⋆ m + 1 | ∞ , H + | u m − Υ δ ⋆ m | ∞ , H + | 2 ⁢ u m + 1 − u m | ∞ , H ≤ L ⁢ ( h 1 + h 2 ) − 1 2 ⁢ ( 2 ⁢ | u m + 1 − Υ δ ⋆ m + 1 | 1 , H + | u m − Υ δ ⋆ m | 1 , H ) + 3 ⁢ max Q ⁡ | u | < 3 ⁢ C δ ⋆ F , 2 ⁢ L ⁢ ( h 1 + h 2 ) − 1 2 ⁢ [ τ 2 ⁢ | ln ⁡ ( τ ) | 2 + ( h 1 2 + h 2 2 ) ⁢ | ln ⁡ ( τ ) | ] + 3 ⁢ δ ⋆ 6 ≤ δ ⋆ 6 + δ ⋆ 2 < δ ⋆ , m = 0 , … , N − 2 ,

which, along with (3.1), yields n δ ⋆ ⁢ ( Υ δ ⋆ 1 / 2 ) = Υ δ ⋆ 1 / 2 and n δ ⋆ ⁢ ( 2 ⁢ Υ δ ⋆ m + 1 − Υ δ ⋆ m ) = 2 ⁢ Υ δ ⋆ m + 1 − Υ δ ⋆ m for m = 0 , … , N − 1 . Thus, from (2.8)–(2.11) and (6.1)–(6.3), we conclude that Υ δ ⋆ m = Υ m for m = 0 , … , N , and the error estimates (6.30) and (6.31) follow as a natural outcome of (6.15) and (6.16), respectively. ∎

7 Numerical Results

We have implemented the proposed numerical methods (LBEFD) and (LBDF2FD) in Python 3.7.0 programs, where we solve the resulting linear systems of algebraic equations using the usual conjugate gradient method, by applying the subroutine cg of the library scipy.sparse.linalg.

When the exact solution to the problem is known, we test the performance of our finite difference methods by computing

the discrete L t ∞ ⁢ ( L x 2 ) -error E 0 ( N , J 1 , J 2 ) := max 0 ≤ n ≤ N ∥ U n − u n ∥ 0 , H ,
the discrete L t ∞ ⁢ ( H x 1 ) -error E 1 ( N , J 1 , J 2 ) := max 0 ≤ n ≤ N | U n − u n | 1 , H and
the discrete L t ∞ ⁢ ( L x ∞ ) -error E ∞ ( N , J 1 , J 2 ) := max 0 ≤ n ≤ N | U n − u n | ∞ , H .

Then, after choosing ν ∈ N , a function f : ( 0 , + ∞ ) ↦ ( 0 , + ∞ ) 3 and ( N , J 1 , J 2 ) = f ⁢ ( ν ) , we compute the experimental order of convergence with respect to 𝜈, corresponding to given values ν 1 and ν 2 of 𝜈, by using the formula ln ⁡ [ E ⁢ ( f ⁢ ( ν 1 ) ) / E ⁢ ( f ⁢ ( ν 2 ) ) ] / ln ⁡ ( ν 2 / ν 1 ) , where E = E 0 , E 1 or E ∞ . In particular, we choose f ⁢ ( ν ) = ( ν , ν , ν ) in the (LBEFD) method and f ⁢ ( ν ) = ( ν , ν , ν ) in the (LBDF2FD) method.

7.1 Example 1

Let T = 1 , D = [ 0 , 1 ] × [ 0 , 1 ] , and let the load 𝑓 be such that the function

u ⁢ ( t , x ) = 1 2 ⁢ exp ⁡ ( 2 + sin ⁡ ( 2 ⁢ π ⁢ t ) ) ⁢ sin ⁡ ( 2 ⁢ π ⁢ x 1 ) ⁢ sin ⁡ ( 2 ⁢ π ⁢ x 2 )

is the exact solution to problem (1.1)–(1.5). The errors we computed are shown in Table 1 and confirm an experimental order of convergence with respect to 𝜈 of first order for the (LBEFD) method and of second for the (LBDF2FD) method.

Table 1

Example 1.

(LBEFD) method					(LBDF2FD) method

𝜈	E 0 ⁢ ( f ⁢ ( ν ) )	Rate	E ∞ ⁢ ( f ⁢ ( ν ) )	Rate	𝜈	E 0 ⁢ ( f ⁢ ( ν ) )	Rate	E 1 ⁢ ( f ⁢ ( ν ) )	Rate
200	7.033(−2)	—	1.393(−1)	—	20	6.304(−2)	—	5.581(−1)	—
400	3.583(−2)	0.97	7.138(−2)	0.96	40	1.677(−2)	1.90	1.489(−1)	1.90
800	1.883(−2)	0.92	3.761(−2)	0.92	80	4.309(−3)	1.96	3.828(−2)	1.95
1600	9.413(−3)	1.00	1.882(−2)	0.99	160	1.089(−3)	1.98	9.684(−3)	1.98
3200	4.880(−3)	0.94	9.769(−3)	0.94	320	2.739(−4)	1.99	2.434(−3)	1.99

7.2 Example 2

Let T = 1 , D = [ 0 , 1 ] × [ 0 , 1 ] , and let the load 𝑓 be such that the function

u ⁢ ( t , x ) = 100 ⁢ e t ⁢ ( x 1 5 + x 2 5 ) ⁢ ∏ i = 1 4 ( x 1 − a i ) ⁢ ( x 2 − a i ) ,

with a 1 = 0 , a 2 = 1 , a 3 = 0.5 and a 4 = 0.25 , is the exact solution to problem (1.1)–(1.5). Computing again the numerical approximation errors, we conclude a first-order experimental order of convergence for the (LBEFD) method and a second-order one for the (LBDF2FD) method, as it is shown in Table 2.

Table 2

Example 2.

(LBEFD) method					(LBDF2FD) method

𝜈	E 0 ⁢ ( f ⁢ ( ν ) )	Rate	E ∞ ⁢ ( f ⁢ ( ν ) )	Rate	𝜈	E 0 ⁢ ( f ⁢ ( ν ) )	Rate	E 1 ⁢ ( f ⁢ ( ν ) )	Rate
200	3.435(−3)	—	1.208(−2)	—	20	1.767(−3)	—	1.547(−2)	—
400	1.766(−3)	0.95	6.336(−3)	0.93	40	4.660(−4)	1.92	4.122(−3)	1.90
800	9.299(−4)	0.92	3.406(−3)	0.89	80	1.195(−4)	1.96	1.060(−3)	1.95
1600	4.659(−4)	0.99	1.708(−3)	0.99	160	3.026(−5)	1.98	2.686(−4)	1.98
3200	2.412(−4)	0.94	8.830(−4)	0.95	320	7.613(−6)	1.99	6.760(−5)	1.99

7.3 Example 3

Let T = 1 , D = [ − 5 , 5 ] × [ − 5 , 5 ] , f = 0 , σ > 0 , and assume the initial condition

u 0 ⁢ ( x ) := exp ⁡ ( − σ ⁢ | x | 2 ) for all ⁢ x ∈ D .

Choosing σ = 1 and setting ψ ⁢ ( t ) := − 4 3 ⁢ ln ⁡ ( 4 − 3 ⁢ e − t ) and a ⁢ ( t ) := 1 4 − 3 ⁢ e − t for t ∈ [ 0 , + ∞ ) , it is easily verified that the function

u GS ⁢ ( t , x ) := exp ⁡ ( ψ ⁢ ( t ) ⁢ e t − a ⁢ ( t ) ⁢ | x | 2 ) for all ⁢ ( t , x ) ∈ [ 0 , + ∞ ) × R 2

is an exact solution to (1.1) with u GS ⁢ ( 0 , x ) = u 0 ⁢ ( x ) for x ∈ R 2 (cf. [1]). Obviously, u GS has almost compact support on 𝖰, and thus we can consider it as the solution to problem (1.1)–(1.5) with initial condition u 0 . In the numerical experiments, we compute approximations of u GS posting the corresponding computational errors in Table 3. We observe, again, a first-order experimental order of convergence for the (LBEFD) method and a second-order one for the (LBDF2FD) method, which seems to be more robust.

Table 3

Example 3.

(LBEFD) method					(LBDF2FD) method

𝜈	E 0 ⁢ ( f ⁢ ( ν ) )	Rate	E ∞ ⁢ ( f ⁢ ( ν ) )	Rate	𝜈	E 0 ⁢ ( f ⁢ ( ν ) )	Rate	E 1 ⁢ ( f ⁢ ( ν ) )	Rate
800	1.003(−3)	—	9.630(−3)	—	40	3.599(−4)	—	8.685(−3)	—
1600	5.002(−4)	1.00	5.092(−3)	0.91	80	8.978(−5)	2.00	2.156(−3)	2.00
3200	2.576(−4)	0.95	2.697(−3)	0.91	160	2.257(−5)	1.99	5.423(−4)	1.99
6400	1.275(−4)	1.01	1.355(−3)	0.99	320	5.665(−6)	1.99	1.361(−4)	1.99

Since 𝑎 and 𝜓 are strictly decreasing functions with

a ⁢ ( 0 ) = 1 , lim t → + ∞ a ⁢ ( t ) = 1 4 , ψ ⁢ ( 0 ) = 0 and lim t → + ∞ ψ ⁢ ( t ) = − 8 3 ⁢ ln ⁡ ( 2 ) ,

we conclude that 0 < u GS ⁢ ( t , x ) ≤ exp ⁡ ( ψ ⁢ ( t ) ⁢ e t ) and u 0 ⁢ ( x ) ≤ exp ⁡ ( − a ⁢ ( t ) ⁢ | x | 2 ) < exp ⁡ ( − 1 4 ⁢ | x | 2 ) for ( t , x ) ∈ [ 0 , + ∞ ) × R 2 , and thus we have

lim t → + ∞ sup x ∈ R 2 u GS ⁢ ( t , x ) = 0 and lim t → + ∞ sup x ∈ R 2 | exp ⁡ ( − a ⁢ ( t ) ⁢ | x | 2 ) − exp ⁡ ( − 1 4 ⁢ | x | 2 ) | = 0 .

Roughly speaking, the initial exponential peak u 0 decays super-exponentially to zero and diffuses slowly to a limit function. This behaviour is visible in the snapshots of the numerical approximations posted in Figure 1, which have been obtained by the (LBDF2FD) method.

Figure 1

Snapshots of the numerical solution of Example 3.

(a)

t = 0

(b)

t = 0.5

(c)

t = 1

(d)

t = 2.5

7.4 Example 4

Let T = 1 , D = [ − 6 , 6 ] × [ − 6 , 6 ] , f = 0 , and assume the initial condition

u 0 ⁢ ( x ) := ∑ j = 1 4 ϵ j ⁢ exp ⁡ ( − 50 ⁢ | x − K j | 2 ) for all ⁢ x ∈ D ,

where K 1 = ( 1 , 0 ) , K 2 = ( 0 , 1 ) , K 3 = ( − 1 , 0 ) , K 4 = ( 0 , − 1 ) and ϵ ∈ R 4 with ϵ j ∈ { − 1 , 1 } for j = 1 , … , 4 . We approximate the solution to the problem for several values of 𝜖, by applying the (LBDF2FD) method with N = 100 and J 1 = J 2 = 500 . In Figures 2, 3 and 4, we post snapshots of the computed solution for ϵ = ( − 1 , 1 , − 1 , 1 ) , ( 1 , 1 , − 1 , − 1 ) , ( 1 , 1 , 1 , 1 ) , respectively. We observe that the initial exponential peaks are diffusing and decaying quickly to zero. Also, it is remarkable that, during the diffusion process, exponential peaks of the same sign tend to merge keeping its sign.

Figure 2

Snapshots of the numerical solution of Example 4 for ϵ = ( − 1 , 1 , − 1 , 1 ) .

(a)

t = 0

(b)

t = 0.1

(c)

t = 0.5

(d)

t = 1

Figure 3

Snapshots of the numerical solution of Example 4 for ϵ = ( 1 , 1 , − 1 , − 1 ) .

(a)

t = 0

(b)

t = 0.1

(c)

t = 0.5

(d)

t = 1

Figure 4

Snapshots of the numerical solution of Example 4 for ϵ = ( 1 , 1 , 1 , 1 ) .

(a)

t = 0

(b)

t = 0.1

(c)

t = 0.5

(d)

t = 1

8 Conclusions

For the approximation of the solution to the semilinear heat equation with logarithmic nonlinearity over a two-dimensional rectangular domain, we propose the (LBEFD) method and the (LBDF2FD) method, which are described in Section 2. The convergence of both numerical methods is established by proving almost optimal order error estimates. Results from numerical experiments for both methods confirm their expected order of convergence and expose their efficiency. Future research plans include the investigation of higher-order numerical methods properly extending the framework developed here.

Acknowledgements

The authors would like to thank the anonymous referees for their useful comments and suggestions. G. E. Zouraris acknowledges the support of The University of Crete via a sabbatical leave (2022–2023).

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Received: 2022-10-27

Revised: 2023-01-24

Accepted: 2023-02-14

Published Online: 2023-03-31

Published in Print: 2023-07-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity

Abstract

1 Introduction

Proof

Proof

2 The Implicit-Explicit Finite Difference Methods

2.1 Preliminaries

Proof

2.2 Formulation of the Numerical Methods

2.2.1 The (LBEFD) Method

2.2.2 The (LBDF2FD) Method

3 Auxiliary Functions

3.1 A 𝛿-Mollifier

Proof

3.2 An 𝜀-Approximation of 𝑔

Proof

4 Consistency Errors

4.1 Time-Discretization Consistency Error

4.2 Space-Discretization Consistency Error

5 Convergence of the (LBEFD) Method

Proof

Proof

6 Convergence of the (LBDF2FD) Method

Proof

Proof

Proof

7 Numerical Results

7.1 Example 1

7.2 Example 2

7.3 Example 3

7.4 Example 4

8 Conclusions

Acknowledgements

References

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