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Spectral collocation method for convection-diffusion equation

  • Jin Li EMAIL logo and Yongling Cheng
Published/Copyright: February 6, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

Spectral collocation method, named linear barycentric rational interpolation collocation method (LBRICM), for convection-diffusion (C-D) equation with constant coefficient is considered. We change the discrete linear equations into the matrix equation. Different from the classical methods to solve the C-D equation, we solve the C-D equation with the time variable and space variable obtained at the same time. Furthermore, the convergence rate of the C-D equation by LBRICM is proved. Numerical examples are presented to test our analysis.

Keywords: linear rational interpolation; convection-diffusion equation; Barycentric form; error estimate; matrix equation
MSC 2010: 65D05; 65L60; 31A30

1 Introduction

In this study, we consider the convection-diffusion (C-D) equation with constant coefficient:

(1) D u ≕ ∂ u ( x , t ) ∂ t + q ∂ u ( x , t ) ∂ x + s ∂ 2 u ( x , t ) ∂ x 2 = f ( x , t ) ,

where ( x , t ) ∈ [ a , b ] × ( 0 , T ] and q and s are the constants related to fluid diffusion coefficient and concentration, respectively.

There are lots of methods to invest the C-D equation, such as finite difference method [1]. Spectral methods [2] have been widely used in lots of scientific engineering area.

In [3], nonlinear C-D equation is studied by the finite-difference lattice Boltzmann model. In reference [4], error estimate of a nonlinear C-D equation by finite element method is presented. In [5], the nonlinear C-D equation was studied and posteriori error estimates were also given. In [6], some spectral and pseudospectral approximations of Jacobi and Legendre type are considered for the C-D equation. In [7], a high-order accurate method for solving the one-dimensional heat and advection-diffusion equations is presented. In reference [8], C-D equation was solved by new formulas of the high-order derivatives of the fifth-kind Chebyshev polynomials. In [9], a transient C-D equation is considered. Time-fractional diffusion equation was solved by Petrov-Galerkin Lucas polynomials in [10]. In reference [11], the time-fractional diffusion equation was dealt by the explicit Chebyshev-Galerkin scheme. In reference [12], fractional diffusion-wave equation was dealt by the shifted fifth-kind Chebyshev polynomials. In reference [13], two-dimensional nonlinear reaction-diffusion equation with Riesz space-fractional was solved by the Legendre-Chebyshev spectral collocation method. In reference [14], fractional calculus approach was used for oxygen diffusion from capillary to tissues.

As for the barycentric interpolation collocation method, there are barycentric Lagrange interpolation collocation method and barycentric rational interpolation collocation method. We can obtain high accuracy by changing the parameter of barycentric rational interpolation polynomial; in the following, we take the barycentric rational interpolation basis function to approximate the unknown function. Similarly, as the finite difference methods, linear barycentric rational method (LBRM) has been studied [15–19] by choosing the parameter in weight function. The LBRM has been used to solve certain problems such as delay Volterra integro-differential equations [20,21], Volterra integral equations [22,23], boundary value problem [24], plane elastic problems [25], incompressible plane elastic problems [26], nonlinear problems [27], and so on [28,29]. Linear barycentric rational collocation method (LBRCM) to solve biharmonic equation [30,31], fractional differential equations [32–34], telegraph equation [35], Volterra integro-differential equation [36], and heat conduction equation [37] are studied.

In this study, we use the linear barycentric rational collocation method to solve the C-D equation. With the help of convergence rate of interpolation barycentric rational function, both the convergence rate of space and time of linear barycentric rational collocation method for the CD equation can be obtained at the same time.

This article is organized as follows: in Section 2, the differentiation matrices and collocation scheme for the C-D equation are presented. In Section 3, the convergence rate of space and time is proved. At last, some numerical examples are listed to illustrate our theorem.

2 Collocation scheme for C-D equation

The area Ω is partitioned into a = x 1 < x 2 < ⋯ < x m = b , 0 = t 1 < t 2 < ⋯ < t n = T with h = b − a m and τ = T n being the uniform partition. However, for the quasi-uniform partition, the second kind of Chebyshev point x = cos ( x i ) , t = cos ( t j ) is chosen.

Let

(2) u ( x , t ) = ∑ j = 1 m L j ( x ) u j ( t ) ,

where

u ( x i , t ) = u i ( t ) , i = 1 , 2 , … , m

and

(3) L j ( x ) = λ j x − x j ∑ k = 1 n λ k x − x k ,

where

λ k = ∑ j ∈ J k ( − 1 ) j ∏ i = j , j ≠ k j + d x 1 x k − x i , J k = { j ∈ { 0 , 1 , … , l − d x } : k − d x ≤ j ≤ k }

is the basis function [36].

Then, we change equation (1) into the following:

(4) ∑ j = 1 m δ i j u ˙ j ( t ) + q ∑ j = 1 m C i j ( 1 ) u j ( t ) + s ∑ j = 1 m C i j ( 2 ) u j ( t ) = f i ( t ) ,

where L j ( x i ) = δ i j , L j ″ ( x i ) = C i j ( 2 ) and f ( x i , t ) = f i ( t ) , i = 1 , 2 , … , m . Then, we have

(5) u ˙ 1 ( t ) ⋮ u ˙ m ( t ) + C 11 ( 1 ) ⋯ C 1 m ( 1 ) ⋮ ⋮ C m 1 ( 1 ) ⋯ C m n ( 1 ) + C 11 ( 2 ) ⋯ C 1 m ( 2 ) ⋮ ⋮ C m 1 ( 2 ) ⋯ C m n ( 2 ) u 1 ( t ) ⋮ u m ( t ) = f 1 ( t ) ⋮ f m ( t ) ,

where

(6) u ( t ) = [ u 1 ( t ) , u 2 ( t ) , … , u m ( t ) ] T , f ( t ) = [ f 1 ( t ) , f 2 ( t ) , … , f m ( t ) ] T .

In order to obtain the time discrete, we take the barycentric interpolation function to the t similarly as x :

(7) u i j = u i ( t j ) = u ( x i , t j ) , i , j = 1 , 2 , … , m , n

and

(8) u i ( t ) = ∑ k = 1 n L k ( t ) u i k ,

then we have

(9) ∑ k = 1 n L ˙ k ( t j ) u 1 k ⋮ ∑ k = 1 n L ˙ k ( t j ) u m k + C 11 ( 1 ) ⋯ C ∣ m ( 1 ) ⋮ ⋮ C m 1 ( 1 ) ⋯ C m n ( 1 ) ∑ k = 1 n L k ( t j ) u 1 k ⋮ ∑ k = 1 n L k ( t j ) u n k + C 11 ( 2 ) ⋯ C ∣ m ( 2 ) ⋮ ⋮ C m 1 ( 2 ) ⋯ C m n ( 2 ) ∑ k = 1 n L k ( t j ) u 1 k ⋮ ∑ k = 1 n L k ( t j ) u n k = f 1 ( t j ) ⋮ f m ( t j ) ,

where j = 1 , 2 , … , n .

We transform the linear equations into the matrix form as:

(10) I n ⊗ D ( 1 ) + q C ( 1 ) ⊗ I n + s C ( 2 ) ⊗ I n U = F

and

(11) LU = F ,

where L = I n ⊗ D ( 1 ) + q C ( 1 ) ⊗ I n + s C ( 2 ) ⊗ I n and ⊗ denotes the Kronecker product [32,38].

3 Convergence and error analysis

We define the error of u ( x ) with r ( x ) as:

(12) e ( x ) ≔ u ( x ) − r ( x ) = ( x − x i ) ⋯ ( x − x i + d ) u [ x i , x i + 1 , … , x i + d , x ]

and

(13) ∣ e ( x ) ∣ = ∑ i = 0 n − d λ i ( x ) [ u ( x ) − p i ( x ) ] ∑ i = 0 n − d λ i ( x ) = A ( x ) B ( x ) = O ( h d + 1 ) ,

where

(14) A ( x ) ≔ ∑ i = 0 n − d ( − 1 ) i u [ x i , … , x i + d , x ]

and

(15) B ( x ) ≔ ∑ i = 0 n − d λ i ( x ) ,

where

(16) λ i ( x ) = ( − 1 ) i ( x − x i ) ⋯ ( x − x i + d )

and

∣ e k ( x ) ∣ ≤ C h d + 1 − k , u ∈ C d + k [ a , b ] , k = 0 , 1 , … .

We first define barycentric rational interpolants as:

(17) r n m ( x , t ) = ∑ i = − d 1 m + d 1 ∑ j = − d 2 n + d 2 w i , j ( x − x i ) ( t − t j ) u ˜ i , j ∑ i = − d 1 m + d 1 ∑ j = − d 2 n + d 2 w i , j ( x − x i ) ( t − t j ) ,

where

(18) w i , j = ( − 1 ) i − d 1 + j − d 2 ∑ k 1 ∈ J i , k 2 ∈ J i , h 1 = k 1 , h ≠ i 1 1 ∣ x i − x h 1 ∣ ∏ h 2 = k 2 , h 2 ≠ j k 2 + d 2 1 ∣ t j − t h 2 ∣

and J i = { k 1 ∈ I m : i − d 1 ≤ k 1 ≤ i } , I m = { 0 , … , m − d 1 } .

The error of u ( x , t ) with r m n ( x , t ) is defined as:

(19) e ( x , t ) ≔ u ( x , t ) − r m n ( x , t ) = ( x − x i ) ⋯ ( x − x i + d 1 ) u [ x i , x i + 1 , … , x i + d 1 , x ] + ( t − t j ) ⋯ ( t − t j + d 2 ) u [ t j , t j + 1 , … , t j + d 2 , t ] .

Now, we present the following theorem.

Theorem 1

For the error e ( x , t ) in (19) and u ( x , t ) , we have

(20) ∣ e ( x , t ) ∣ ≤ C ( h d 1 + 1 + τ d 2 + 1 ) .

Proof

For ( x , t ) ∈ Ω and e ( x , t ) , we have

(21) e ( x , t ) = u ( x , t ) − r m n ( x , t ) = ∑ i = 0 m − d 1 ∑ j = 0 n − d 2 λ i ( x ) λ j ( t ) ( u ( x , t ) − r m n ( x , t ) ) ∑ i = 0 m − d 1 ∑ j = 0 n − d 2 λ i ( x ) λ j ( t )

and

(22) u ( x , t ) − r m n ( x , t ) = u ( x , t ) − u ( x m , t ) + u ( x m , t ) − r m n ( x , t ) = ( x − x i ) ⋯ ( x − x i + d 1 ) u [ x i , x i + 1 , … , x i + d 1 , x m , t ] + ( t − t j ) ⋯ ( t − t j + d 2 ) u [ t j , t j + 1 , … , t j + d 2 , x , t ] .

We reach that

(23) u ( x , t ) − r m n ( x , t ) = ∑ i = 0 m − d 1 ( − 1 ) i u [ x i , x i + 1 , … , x i + d 1 , x m , t ] ∑ i = 0 m − d 1 λ i ( x ) + ∑ j = 0 n − d 2 ( − 1 ) j u [ t j , t j + 1 , … , t j + d 2 , x , t ] ∑ j = 0 n − d 2 λ j ( t ) .

By the similarly analysis in Floater and Kai [29], we have

(24) ∑ i = 0 m − d 1 λ i ( x ) ≥ 1 d 1 ! h d 1 + 1

and

(25) ∑ j = 0 n − d 2 λ j ( t ) ≥ 1 d 2 ! τ d 2 + 1 .

Combining (23), (24), and (25) together, the proof of Theorem 1 is completed.□

Corollary 1

For the e ( x , t ) defined in (19), we have

(26) ∣ e x ( x , t ) ∣ ≤ C ( h d 1 + τ d 2 + 1 ) , u ( x , t ) ∈ Ω , ∣ e t ( x , t ) ∣ ≤ C ( h d 1 + 1 + τ d 2 ) , u ( x , t ) ∈ Ω , ∣ e x x ( x , t ) ∣ ≤ C ( h d 1 − 1 + τ d 2 + 1 ) , u ( x , t ) ∈ Ω .

Let u ( x , t ) be the analysis solution of (1), u ( x m , t n ) is its approximate value, there holds

(27) lim m , n → ∞ u ( x m , t n ) = u ( x , t ) ,

(28) D u ( x m , t n ) = f ( x , t ) ,

and

(29) lim m , n → ∞ D u ( x m , t n ) = f ( x , t ) .

Based on the aforementioned lemma, we obtain the following theorem.

Theorem 2

Let operator D be defined as (28) and f ( x , t ) ∈ Ω , and we have

∣ u ( x , t ) − u ( x m , t n ) ∣ ≤ C ( h d 1 − 1 + τ d 2 ) .

Proof

As

(30) D u ( x , t ) − D u ( x m , t n ) = u t ( x , t ) − u t ( x m , t n ) + q [ u x ( x , t ) − u x ( x m , t n ) ] + s [ u x x ( x , t ) − u x x ( x m , t n ) ] ≔ R 1 ( x , t ) + R 2 ( x , t ) + R 3 ( x , t ) ,

where

R 1 ( x , t ) = u t ( x , t ) − u t ( x m , t n ) , R 2 ( x , t ) = q [ u x ( x , t ) − u x ( x m , t n ) ] , R 3 ( x , t ) = s [ u x x ( x , t ) − u x x ( x m , t n ) ] .

As for the R 1 , we have

(31) R 1 ( x , t ) = u t ( x , t ) − u t ( x m , t n ) = u t ( x , t ) − u t ( x m , t ) + u t ( x m , t ) − u t ( x m , t n ) = ∑ i = 0 m − d 1 ( − 1 ) i u t [ x i , x i + 1 , … , x i + d 1 , x , t ] ∑ i = 0 m − d 1 λ i ( x ) + ∑ j = 0 n − d 2 ( − 1 ) j u t [ t j , t j + 1 , … , t j + d 2 , x m , t ] ∑ j = 0 n − d 2 λ j ( t ) = e t ( x , t ) + e t ( x m , t ) .

By the corollary, we obtain

(32) ∣ R 1 ( x , t ) ∣ ≤ ∣ e t ( x , t ) + e t ( x m , t ) ∣ ≤ C ( h d 1 + 1 + τ d 2 ) .

Similarly, for R 2 ( x , t ) and R 3 ( x , t ) , we have

(33) R 2 ( x , t ) = u x ( x , t ) − u x ( x m , t n ) = e x ( x , t ) + e x ( x m , t ) ,

(34) ∣ R 2 ( x , t ) ∣ ≤ ∣ e x ( x , t ) + e x ( x m , t ) ∣ ≤ C ( h d 1 + τ d 2 + 1 ) ,

and

(35) R 3 ( x , t ) = u x x ( x , t ) − u x x ( x m , t n ) = e x x ( x , t n ) + e x x ( x m , t n ) ,

(36) ∣ R 3 ( x , t ) ∣ ≤ ∣ e x x ( x , t ) + e x x ( x m , t ) ∣ ≤ C ( h d 1 − 1 + τ d 2 + 1 ) .

Combining the identity (30), (32), (34), and (36), the proof of theorem is complete.□

4 Numerical examples

Three examples are presented to illustrate our theorem. All examples were performed on personal computer by Matlab 2013a with a (configuration: Intel(R) Core(TM) i5-8265U CPU @ 1.60GHz 1.80 GHz).

Example 1

Consider the C-D equation:

u t ( x , t ) = u x x ( x , t ) − u x ( x , t ) + f b 0 ( x , t ) ,

with the condition

u ( x , 0 ) = 0 , u x ( 1 , t ) + b 0 u ( 1 , t ) = 0 , u x ( 0 , t ) − u ( 0 , t ) = 0 .

Its analysis solutions is

u ( x , t ) = [ 3 e x − e ( 1 + x + x 2 ) ] e − δ ( b 0 ) t ( 1 − e − t ) .

In this example, we test the linear barycentric rational with the equidistant nodes. Table 1 shows that the convergence rate is O ( h d 1 ) with d 2 = 9 first given for the space area for t = 1 . In Table 2, for the space area partition d 1 = 9 first given, the convergence rate of times is O ( τ d 2 ) , which agrees with our theorem analysis.

Table 1

Convergence rate of equidistant nodes with d 2 = 9 and t = 1

m , n d 1 = 1 h α d 1 = 2 h α d 1 = 3 h α d 1 = 4 h α
10 × 10 3.5548 × 1 0 ‒ 3 3.9350 × 1 0 ‒ 4 1.1116 × 1 0 ‒ 4 1.0400 × 1 0 ‒ 5
20 × 20 1.4005 × 1 0 ‒ 3 1.3438 8.0420 × 1 0 ‒ 5 2.2907 1.0650 × 1 0 ‒ 5 3.3837 5.8204 × 1 0 ‒ 7 4.1593
40 × 40 5.3926 × 1 0 ‒ 4 1.3769 1.5825 × 1 0 ‒ 5 2.3453 1.0249 × 1 0 ‒ 6 3.3774 2.9676 × 1 0 ‒ 8 4.2937
80 × 80 2.0200 × 1 0 ‒ 4 1.4166 3.0026 × 1 0 ‒ 6 2.3979 9.6236 × 1 0 ‒ 8 3.4127 1.4282 × 1 0 ‒ 9 4.3770
Table 2

Convergence rate of equidistant nodes with d 1 = 9 and t = 1

m , n d 2 = 1 h α d 2 = 2 h α d 2 = 3 h α d 2 = 4 h α
10 × 10 1.0353 × 1 0 ‒ 2 2.1124 × 1 0 ‒ 3 6.3532 × 1 0 ‒ 4 1.7118 × 1 0 ‒ 5
20 × 20 4.7875 × 1 0 ‒ 3 1.1127 5.1680 × 1 0 ‒ 4 2.0312 7.0607 × 1 0 ‒ 5 3.1696 1.1376 × 1 0 ‒ 6 3.9115
40 × 40 2.3078 × 1 0 ‒ 3 1.0528 1.2764 × 1 0 ‒ 4 2.0176 8.3487 × 1 0 ‒ 6 3.0802 7.1789 × 1 0 ‒ 8 3.9860
80 × 80 1.1346 × 1 0 ‒ 3 1.0244 3.1707 × 1 0 ‒ 5 2.0092 1.0157 × 1 0 ‒ 6 3.0391 4.4931 × 1 0 ‒ 9 3.9980

We choose m = 40 , n = 40 , d 1 = 9 , d 2 = 9 , and β = 1 to test our algorithm. Figure 1 shows the error estimate of equidistant nodes, and Figure 2 shows the error estimate of Chebyshev nodes. From Figures 1 and 2, we know that the barycentric rational interpolation collocation method has higher accuracy under the condition of Chebyshev nodes, and the barycentric rational interpolation collocation method also has higher accuracy under the condition of equidistant nodes.

Figure 1 Error estimate of equidistant nodes with m = 40 m=40 , n = 40 n=40 , d 1 = 9 {d}_{1}=9 , d 2 = 9 {d}_{2}=9 , and β = 1 \beta =1 .
Figure 1

Error estimate of equidistant nodes with m = 40 , n = 40 , d 1 = 9 , d 2 = 9 , and β = 1 .

Figure 2 Error estimate of Chebyshev nodes with m = 40 m=40 , n = 40 n=40 , d 1 = 9 {d}_{1}=9 , d 2 = 9 {d}_{2}=9 , and β = 1 \beta =1 .
Figure 2

Error estimate of Chebyshev nodes with m = 40 , n = 40 , d 1 = 9 , d 2 = 9 , and β = 1 .

In Table 3, we test the linear barycentric rational with the Chebyshev nodes, which that shows the convergence rate is O ( h d 2 + 1 ) with d 1 = 9 first given for the space area. In Table 4, for d 2 = 9 first given, the convergence rate of times is O ( τ d 1 + 1 ) , which agrees with our theorem analysis.

Table 3

Convergence rate of Chebyshev nodes with d 1 = 9 and t = 1

m , n d 2 = 1 h α d 2 = 2 h α d 2 = 3 h α d 2 = 4 h α
10 × 10 1.1867 × 1 0 ‒ 3 9.1568 × 1 0 ‒ 5 1.7898 × 1 0 ‒ 5 1.1576 × 1 0 ‒ 6
20 × 20 2.6617 × 1 0 ‒ 4 2.1565 1.0834 × 1 0 ‒ 5 3.0792 1.0571 × 1 0 ‒ 6 4.0816 4.9817 × 1 0 ‒ 8 4.5383
40 × 40 6.0449 × 1 0 ‒ 5 2.1386 1.2475 × 1 0 ‒ 6 3.1185 5.9525 × 1 0 ‒ 8 4.1505 1.4384 × 1 0 ‒ 9 5.1141
80 × 80 1.4033 × 1 0 ‒ 5 2.1069 1.4806 × 1 0 ‒ 7 3.0748 3.5266 × 1 0 ‒ 9 4.0771 8.8800 × 1 0 ‒ 9 —
Table 4

Convergence rate of Chebyshev nodes with d 2 = 9 and t = 1

n d 1 = 1 h α d 1 = 2 h α d 1 = 3 h α d 1 = 4 h α
10 × 10 2.4674 × 1 0 ‒ 3 7.7254 × 1 0 ‒ 5 8.2887 × 1 0 ‒ 6 1.0909 × 1 0 ‒ 6
20 × 20 1.0512 × 1 0 ‒ 3 1.2309 1.8978 × 1 0 ‒ 5 2.0253 6.8668 × 1 0 ‒ 7 3.5934 1.7571 × 1 0 ‒ 9 9.2781
40 × 40 3.7121 × 1 0 ‒ 4 1.5017 1.8696 × 1 0 ‒ 6 3.3435 1.8229 × 1 0 ‒ 8 5.2353 3.7351 × 1 0 ‒ 11 5.5559
80 × 80 1.2165 × 1 0 ‒ 4 1.6095 6.5999 × 1 0 ‒ 6 — 7.6684 × 1 0 ‒ 6 — 4.4403 × 1 0 ‒ 6 —

In Table 5, we show that the error estimate of time equals t = 1 , 10 , 40 , 80 , and 100 with m = 40 , n = 40 , d 1 = 9 , d 2 = 9 , and β = 1 , and the error estimate is given as blow for the equidistant nodes and Chebyshev nodes.

Table 5

Convergence rate of long time with m = 40 , n = 40 , d 1 = 9 , d 2 = 9 , and β = 1

n t = 1 t = 10 t = 40 t = 80 t = 100
Equidistant nodes 4.2413 × 1 0 ‒ 11 3.4452 × 1 0 ‒ 6 3.8497 × 1 0 ‒ 3 6.4714 × 1 0 ‒ 3 5.0873 × 1 0 ‒ 3
Chebyshev nodes 1.5197 × 1 0 ‒ 10 1.2543 × 1 0 ‒ 7 3.6741 × 1 0 ‒ 3 1.3245 × 1 0 ‒ 01 3.1104 × 1 0 ‒ 01

Example 2

Consider the C-D equation:

u t ( x , t ) = u x x ( x , t ) − u x ( x , t ) + f ( x , t ) , 0 < x < 1 ; t > 0 u ( x , 0 ) = 0 , 0 < x < 1 u x ( 1 , t ) + 2 u ( 1 , t ) = 0 , u x ( 0 , t ) − 3 2 u ( 0 , t ) = 0 , t > 0 .

Its analysis solutions is

u ( x , t ) = 1 + 1 2 x − x 2 e x t e − β t

and

f ( x , t ) = 1 2 e x e − β ( 2 + x − 2 x 2 − 2 β t − β x t + 2 β x 2 t + 3 t + 4 x t ) .

We still choose m = 40 , n = 40 , d 1 = 9 , d 2 = 9 , and β = 1 to test our theorem. Figure 3 shows the error estimate of equidistant nodes, and Figure 4 shows the error figure of Chebyshev nodes. From Figures 3 and 4, we know that the barycentric rational interpolation collocation method has higher accuracy under the condition of Chebyshev nodes, and the barycentric rational interpolation collocation method also has higher accuracy under the condition of equidistant nodes.

Figure 3 Convergence rate of equidistant nodes with m = 40 m=40 , n = 40 n=40 , d 1 = 9 {d}_{1}=9 , d 2 = 9 {d}_{2}=9 , and β = 0.6 \beta =0.6 .
Figure 3

Convergence rate of equidistant nodes with m = 40 , n = 40 , d 1 = 9 , d 2 = 9 , and β = 0.6 .

Figure 4 Error estimate of Chebyshev nodes with m = 40 m=40 , n = 40 n=40 , d 1 = 9 {d}_{1}=9 , d 2 = 9 {d}_{2}=9 , and β = 0.6 \beta =0.6 .
Figure 4

Error estimate of Chebyshev nodes with m = 40 , n = 40 , d 1 = 9 , d 2 = 9 , and β = 0.6 .

In this example, in order to test our theorem analysis, we have first given d 1 = 9 and t = 1 to be the exact solution. Table 6 shows that the convergence rate of equidistant nodes is O ( τ d 2 ) , which agrees with our theorem analysis. In Table 7, in order to test the convergence rate of space, d 2 = 9 is first given to be the exact solution, and the convergence rate of times is O ( h d 1 ) , which is one order higher than our theorem analysis.

Table 6

Convergence rate of equidistant nodes with d 1 = 9 and t = 1

m , n d 2 = 1 h α d 2 = 2 h α d 2 = 3 h α d 2 = 4 h α
10 × 10 2.0117 × 1 0 ‒ 2 9.9996 × 1 0 ‒ 4 2.6462 × 1 0 ‒ 4 9.8527 × 1 0 ‒ 6
20 × 20 7.6706 × 1 0 ‒ 3 1.3910 1.9840 × 1 0 ‒ 4 2.3334 2.4016 × 1 0 ‒ 5 3.4619 5.3162 × 1 0 ‒ 7 4.2120
40 × 40 2.8333 × 1 0 ‒ 3 1.4368 3.7231 × 1 0 ‒ 5 2.4139 2.1654 × 1 0 ‒ 6 3.4713 2.5515 × 1 0 ‒ 8 4.3810
80 × 80 1.0279 × 1 0 ‒ 3 1.4628 6.7990 × 1 0 ‒ 6 2.4531 1.9398 × 1 0 ‒ 7 3.4807 2.0626 × 1 0 ‒ 9 3.6288
Table 7

Convergence rate of equidistant nodes with d 2 = 9 and t = 1

m , n d 1 = 1 h α d 1 = 2 h α d 1 = 3 h α d 1 = 4 h α
10 × 10 2.6369 × 1 0 ‒ 2 3.4104 × 1 0 ‒ 3 1.0652 × 1 0 ‒ 3 6.8651 × 1 0 ‒ 5
20 × 20 6.0571 × 1 0 ‒ 3 1.0630 3.9190 × 1 0 ‒ 4 2.0623 5.6503 × 1 0 ‒ 5 3.1775 2.1560 × 1 0 ‒ 6 3.9337
40 × 40 1.4660 × 1 0 ‒ 3 1.0205 4.7384 × 1 0 ‒ 5 2.0218 3.2950 × 1 0 ‒ 6 3.0737 6.6715 × 1 0 ‒ 8 3.9880
80 × 80 3.5956 × 1 0 ‒ 4 1.0149 5.8076 × 1 0 ‒ 6 2.0157 1.9852 × 1 0 ‒ 7 3.0402 2.0590 × 1 0 ‒ 9 4.0053

In Table 8, we test the linear barycentric rational with the Chebyshev nodes, and shows that the convergence rate is O ( h d 1 + 1 ) with d 2 = 9 and t = 10 first given for the space area. In Table 9, for the space area partition d 1 = 9 first given, the convergence rate of times is O ( τ d 2 + 1 ) , which is one order higher than our theorem analysis.

Table 8

Convergence rate of Chebyshev nodes with d 2 = 9 and t = 10

m , n d 1 = 1 h α d 1 = 2 h α d 1 = 3 h α d 1 = 4 h α
10 × 10 3.5254 × 1 0 ‒ 2 1.4203 × 1 0 ‒ 3 1.5504 × 1 0 ‒ 3 1.5309 × 1 0 ‒ 3
20 × 20 1.2567 × 1 0 ‒ 2 1.4882 1.0407 × 1 0 ‒ 4 3.7706 6.3260 × 1 0 ‒ 6 7.9371 5.5968 × 1 0 ‒ 6 8.0956
40 × 40 3.9582 × 1 0 ‒ 3 1.6667 9.8234 × 1 0 ‒ 6 3.4051 1.0552 × 1 0 ‒ 7 5.9057 2.0826 × 1 0 ‒ 8 8.0701
80 × 80 1.7563 × 1 0 ‒ 3 1.1723 1.0063 × 1 0 ‒ 6 3.2871 3.5781 × 1 0 ‒ 7 — 1.2045 × 1 0 ‒ 6 —
Table 9

Convergence rate of Chebyshev nodes with d 1 = 9 and t = 10

m , n d 2 = 1 h α d 2 = 2 h α d 2 = 3 h α d 2 = 4 h α
10 × 10 5.2543 × 1 0 ‒ 2 3.7959 × 1 0 ‒ 2 2.4572 × 1 0 ‒ 2 1.2281 × 1 0 ‒ 2
20 × 20 1.8911 × 1 0 ‒ 2 1.4743 7.4472 × 1 0 ‒ 3 2.3497 3.4116 × 1 0 ‒ 3 2.8485 1.4325 × 1 0 ‒ 3 3.0998
40 × 40 4.7752 × 1 0 ‒ 3 1.9856 1.0058 × 1 0 ‒ 3 2.8884 2.4478 × 1 0 ‒ 4 3.8009 5.6804 × 1 0 ‒ 5 4.6564
80 × 80 1.1944 × 1 0 ‒ 3 1.9993 1.2625 × 1 0 ‒ 4 2.9940 1.5335 × 1 0 ‒ 5 3.9966 2.3166 × 1 0 ‒ 6 4.6159

In Table 10, we show that the error estimate of time equals t = 1 , 10 , 40 , 80 , and 100 with m = 40 , n = 40 , d 1 = 9 , d 2 = 9 , and β = 1 , the error estimate is given for the equidistant nodes and Chebyshev nodes. It is shown that linear barycentric rational collocation method is still effective for the long time.

Table 10

Convergence rate of long time with m = 40 , n = 40 , d 1 = 9 , d 2 = 9 , and β = 1

n t = 1 t = 10 t = 40 t = 80 t = 100
Equidistant nodes 1.1725 × 1 0 ‒ 9 8.0311 × 1 0 ‒ 9 1.2664 × 1 0 ‒ 4 3.2401 × 1 0 ‒ 3 7.0226 × 1 0 ‒ 3
Chebyshev nodes 3.8538 × 1 0 ‒ 8 1.4320 × 1 0 ‒ 9 3.7099 × 1 0 ‒ 5 5.5803 × 1 0 ‒ 3 2.3399 × 1 0 ‒ 2

Example 3

Consider the nonlinear C-D equation:

u t ( x , t ) = ( u + u 2 ) u x x ( x , t ) + ( 2 u + 1 ) u x 2 ( x , t ) − 2 u u x ( x , t ) + 2 u ( 1 − 2 u − 6 u 2 ) , 0 < x < 1 ; t > 0 u ( x , 0 ) = e 2 x , u ( x , 1 ) = e 2 x + 2 .

Its analysis solutions is

u ( x , t ) = e 2 x + 2 t ,

and the boundary condition and initial condition are also given by the analysis solution.

In this example, we take direct linearization scheme with u 0 :

u t ( x , t ) = ( u 0 + u 0 2 ) u x x ( x , t ) + ( 2 u 0 + 1 ) u x 2 ( x , t ) − 2 u 0 u x ( x , t ) + 2 u ( 1 − 2 u 0 − 6 u 0 2 )

and we test our algorithm to solve the nonlinear C-D equation. In order to test our theorem analysis, we have first given d 1 = 9 and t = 1 to be the exact solution. Table 11 shows that the convergence rate of equidistant nodes is O ( τ d 2 ) , which agrees with our theorem analysis. In Table 12, in order to test the convergence rate of space, d 2 = 9 is first given to be the exact solution and the convergence rate of times is O ( h d 1 + 1 ) , which is one order higher than our theorem analysis. From this example, the same convergence rate is the same as the linear C-D equation, while the theorem analysis is beyond our goal and will be given in another study.

Table 11

Convergence rate of equidistant nodes with d 1 = 9 and t = 1

m , n d 2 = 1 h α d 2 = 2 h α d 2 = 3 h α d 2 = 4 h α
10 × 10 8.0557 × 1 0 ‒ 4 1.2053 × 1 0 ‒ 5 3.0259 × 1 0 ‒ 6 1.6513 × 1 0 ‒ 7
20 × 20 4.1120 × 1 0 ‒ 4 0.9702 3.4240 × 1 0 ‒ 6 1.8156 4.1782 × 1 0 ‒ 7 2.8564 1.4356 × 1 0 ‒ 8 3.5239
40 × 40 1.7850 × 1 0 ‒ 4 1.2039 8.0203 × 1 0 ‒ 7 2.0940 4.8292 × 1 0 ‒ 8 3.1130 9.0948 × 1 0 ‒ 10 3.9805
80 × 80 7.0234 × 1 0 ‒ 5 1.3457 1.6862 × 1 0 ‒ 7 2.2499 4.8670 × 1 0 ‒ 9 3.3107 1.4183 × 1 0 ‒ 9 —
Table 12

Convergence rate of equidistant nodes with d 2 = 9 and t = 1

m , n d 1 = 1 h α d 1 = 2 h α d 1 = 3 h α d 1 = 4 h α
10 × 10 3.9671 × 1 0 ‒ 4 7.6555 × 1 0 ‒ 6 2.4860 × 1 0 ‒ 6 1.9775 × 1 0 ‒ 7
20 × 20 1.2339 × 1 0 ‒ 4 1.6848 9.9756 × 1 0 ‒ 7 2.9400 1.5130 × 1 0 ‒ 7 4.0383 7.5609 × 1 0 ‒ 9 4.7090
40 × 40 3.7144 × 1 0 ‒ 5 1.7320 1.2982 × 1 0 ‒ 7 2.9419 9.5794 × 1 0 ‒ 9 3.9813 2.8567 × 1 0 ‒ 10 4.7261
80 × 80 1.0934 × 1 0 ‒ 5 1.7643 1.6644 × 1 0 ‒ 8 2.9634 7.0405 × 1 0 ‒ 10 3.7662 8.2156 × 1 0 ‒ 10 —

Figure 5 shows the error estimate of equidistant nodes. From Figure 5, we know that the barycentric rational interpolation collocation method has higher accuracy under the condition of equidistant nodes.

Figure 5 Convergence rate of equidistant nodes with m = 20 m=20 , n = 20 n=20 , d 1 = 9 {d}_{1}=9 , d 2 = 9 {d}_{2}=9 , and t = 1 t=1 .
Figure 5

Convergence rate of equidistant nodes with m = 20 , n = 20 , d 1 = 9 , d 2 = 9 , and t = 1 .

5 Conclusion

In this study, (1+1)-dimensional C-D equation has been solved by LBRICM, and the discrete C-D equation can be written to matrix equation using the Kronecker product. For the linear C-D equation, the convergence rate of space and time at the same time is proved with the constant coefficient. However, for the nonlinear C-D equation, the convergence rate is the same. As for (2+1)-dimensional linear and nonlinear C-D equation, magneto-micropolar equations, and fractional C-D equation, we will study in further article to overcome the difficulty of the full coefficient matrix.

Acknowledgements

The work of Jin Li was supported by Natural Science Foundation of Shandong Province (Grant No. ZR2022MA003). The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

  1. Conflict of interest: Authors declare that they have no conflicts of interest.

  2. Data availability statement: No data is associated with this research.

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Received: 2022-10-28
Revised: 2023-06-28
Accepted: 2023-09-17
Published Online: 2024-02-06

© 2024 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/dema-2023-0110
Keywords for this article
linear rational interpolation; convection-diffusion equation; Barycentric form; error estimate; matrix equation
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Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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