an algorithm based on dqm with modified trigonometric cubic b-

* Corresponding Author. Email address: [email protected], Tel: +91 8126743980

21

An algorithm based on DQM with modified trigonometric cubic B-

splines for solving coupled viscous Burgers’ equations

Brajesh Kumar Singh1*, Pramod Kumar1

(1) Department of Mathematics, School of Physical and Decision Sciences, Babasaheb Bhimrao Ambedkar

University Lucknow 226 025 INDIA

Copyright 2018 © Brajesh Kumar Singh and Pramod Kumar. This is an open access article distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

Abstract

This paper presents a new algorithm: “modified trigonometric cubic B-spline differential quadrature

method (MTB-DQM)” for solving time dependent partial differential equations. Specially, MTB-DQM

with time integration algorithm has been employed for numeric study of Burgers’ equation. The MTB-

DQM (DQM with modified trigonometric cubic B-splines as new basis) is used to reduce the initial

boundary value system of Burgers’ equation into an initial value system of ordinary differential equations

(ODEs), which is solved via SSP-RK54 algorithm. The accuracy and efficiency of the method is illustrated

by four test problems in terms of 2 ,L L errors and their comparisons with existing results. The proposed

scheme is shown conditionally stable via matrix stability analysis method for various grid points. The

computed results are better than the results due to almost all existing schemes.

Keywords: Modified trigonometric cubic B-spline DQM; Burgers’ equation; SSP-RK54; Matrix stability analysis

method, Thomas algorithm

1 Introduction

The Burgers’ equation, Navier–Stoke’s equation without the stress term, is the easiest nonlinear physical

model various flows problems consisting of hydrodynamic turbulence, sound/shock wave theory, vorticity

transportation, dispersion in porous media, modeling of turbulent fluid, continuous stochastic processes.

This equation was first introduced by Bateman [1], for details study, we refer readers to [2-6]. We consider

initial value system of one dimensional (1D) coupled viscous Burgers’ equation:

2

2

2

12

, ( ,0) ( )

, ( ,0) ( ), , t 0.

uvu u uu u x x

t x x x

uvv v vv v x x x

t x x x

(1.1)

Communications in Numerical Analysis 2018 No.1 (2018) 21-41

Available online at www.ispacs.com/cna

Volume 2018, Issue 1, Year 2018 Article ID cna-00333, 21 Pages

doi:10.5899/2018/cna-00333

Research Article

Communications in Numerical Analysis 2018 No. 1 (2018) 21-41 22

http://www.ispacs.com/journals/cna/2018/cna-00333/

International Scientific Publications and Consulting Services

with Dirichlet boundary conditions

1 2 3 4( , ) ( ), ( , ) ( ), ( , ) ( ), ( , ) ( ), 0,u a t g t u b t g t v a t g t v b t g t t (1.2)

and 2D nonlinear coupled viscous Burger’s equation:

2 2

2 2

2 2

2 2

, , ,0 ,

, , ,0 , , , , 0

u u u u uu v u x y x y

t x y x y

v v v v vu v v x y x y x y t

t x y x y

(1.3)

with Dirichlet boundary conditions as follows:

, , , , and , , , , , , , 0,u x y t x y t v x y t x y t x y t (1.4)

Where 1 [ , ]a b , , : , x y a x b c x d computational domains in one and two

dimension, and is the boundary of , , constants and , arbitrary constants which depend

upon the system parameters, eg., Peclet number, stokes velocity of particles and Brownian diffusivity

[7]. ,u v components of velocity profile,

2 2

2 2

u u

x y

diffusion term and coefficient of

viscosity ( 0) and , ,1 2 3 4, , ,g g g g , and are known functions.

In the past years, a lot of efforts have been made for the computation of accuracy and efficiency of various

schemes for Burgers’ equation with different values of kinematic viscosity. Burgers’ equation has been

solved using many approaches, among others, Hofe Cole transform method [5-6], finite element method

[8], finite difference method [9,32], implicit finite difference method [10,30], compact finite difference

method [11-13], Fourier Pseudospectral method [14], variational iteration method [15], collocation

schemes with different B-splines such as Quartic B-spline [31], cubic B-splines [16-17], modified cubic B-

splines [18], modified trigonometric cubic B-spline [19], extended B-spline [20], reproducing kernel

method [21], quadratic B-spline finite elements [22]. For more schemes on (2+1)D Buregers’ equation, the

interested readers are referred to [57-62]. (1+1)D coupled viscous Burgers’equation has been solved

numerically using various techniques, among others, lattice Boltzmann method [23], differential

quadrature method [24], fully implicit finite difference method [25], composite finite difference and Haar

wavelets schemes [26], discrete adomian decomposition method [27], variational iteration method [28],

cubic spline techniques [29] and many more.

Differential quadrature method (DQM) developed by Bellman et al. [33], widely used for numeric study of

partial differential equations (PDEs). After the seminal paper of Bellman and Quan and Chang [34-35],

DQM has been implemented with different set of base functions, e.g., cubic B-spline DQM [36], DQM

based on Fourier expansion and Harmonic function [37-38], sinc DQM [39], generalized DQM [40] and

MCB-DQM [41-44], DQM with modified extended cubic B-splines (mECDQ) [45], Polynomial based

DQM [53], quartic B-spline based DQM [52,54], Quartic and quintic B-spline methods [55], exponential

cubic B-spline DQM [56].

The main goal of this paper is to develop a new method called “modified trigonometric cubic-B-spline

DQM (MTB-DQM)” for solving time dependent PDEs. Specially, Burgers’ equation has been studied

numerically by employing MTB-DQM with time integration algorithm.

The paper is organized into five more sections. Specifically, Section 2 deals with the description of MTB-

DQM. Section 3 is devoted to the procedure for the implementation of method for the 1D and 2D coupled

viscous Burgers’ equations. The stability analysis of MTB-DQM for nonlinear coupled viscous Burgers’

equation is presented in Section 4. Section 5 is concerned with four test problems with the main aim to

establish the accuracy of the proposed method in terms of 2 ,L L errors and their comparisons with




existing results for four test problems. Section 6 concludes the paper with reference to critical analysis and

research perspectives.

2 Modified trigonometric cubic B-spline differential quadrature method

This section deals with the description of MTB-DQM for Burgers’ equation. The weighting coefficients

being depends on only the grid spacing. The domains 1and defined by

2

1 : , an , : , d x a x b x y a x b c y d

Are portioned uniformly in each direction with the following knots:

1 2 1 1 2 1, ; , ,x x y yi N N j N Na x x x x x b c y y y y y d

with space step in x , y directions denoted by ;1 1

x y

x y

b a d ch h

N N

respectively. Let

1, ,i i jx x y be the generic grids and : , ; : , , .i i i i j i j i ju u t u x t u u t u x y t

The trigonometric cubic B-spline function [19] is defined as follows

3

1

2

2 3 1 4 1 1 2

2

4 1 3 4 2 3 2 3

3

4 3 4

( ), [ , )

( )( ( ) ( ) ( ) ( )) ( ) ( ), [ , )1

( )( ( ) ( ) ( ) ( )) ( ) ( ), [ , )

( ), [ , )

m m m

m m m m m m m m m

m

m m m m m m m m m

m m m

p x x x x

p x p x q x q x p x q x p x x x xx

q x p x q x q x p x p x q x x x x

q x x x x

(2.5)

where 3

( ) sin , ( ) sin , sin sin sin2 2 2 2

m m x xm m x

x x x x h hp x q x h

.

where the set 0 1 1{ , , , , }

x xN NT T T T of trigonometric cubic B-spline functions form a basis for the

computational region 1 . Setting

2

1 2 4 3

2

5 622

sin2 32

, , ,3 31 2cos( )

sin( )sin 4sin2 2

3cos3 9cos( ) 2

a ,3

sin 2 4cos( )16sin 2cos cos22 2 2

x

x xxx

x

x

xx x xx

h

a a a ah hh

h

h

ha

hh h hh

The values of the trigonometric B-spline :i j i jT T x and its first and second derivatives ' ': ,ij i jT T x

'' '':ij i jT T x at jx is, respectively, read as:

4 62

' ''

3 51

, 1 , 0, 0

, , , ,, 1 1 1

otherwise otherwise otherwise0, 0, 0,

i j ij ij

a i j a i ja i j

T T a T aa i j i j i j

(2.6)

The modified trigonometric cubic B-splines are defined in similar manner as in [41, 45]:

ℝ ℝ




1 1 0

2 2 0

1 1 1

1

2 ,

, 3,..., - 2 ,

,

2

m m x

N N N

N N N

x T x T x

x T x T x

x T x m N

x T x T x

x T x T x

(2.7)

Where 1 2, , ...,xN forms a basis of

1 . The r-th order partial derivatives of ,u x t with respect

to x approximated at ix read:

1

,i k x

r Nri

k Nrk

ua u i

x

(2.8)

The r-th order partial derivatives of ( , , )u x y t with respect to ,x y at the grid point ( , )i jx y can be

computed as follows:

1 1

1 1

, , , ,

,

, , , ,

i k x i k x

j k y j k y

r rN Nr r

i j kj N i j kj Nr rk k

r rM Mr r

i j ik N i j ik Nr rk k

u vx y a u i x y a v i

x x

u vx y b u j x y b v j

y y

(2.9)

i

r

ja and

i

r

jb 1,2r are the weighting coefficients of the r-th order partial derivatives with respect to

x and y .

2.1. Computation of the weighting coefficients

In order to evaluate the weighting coefficients (1)

ija of Eq. (2.8), the modified trigonometric cubic B-

spline ,xm Nx m are used. Setting ' ': , : . mi m i mi m ix x Accordingly, the approximation

for the first order spatial derivative is given by:

(1)

1

, , .x

x

N

mi i m Na i m

(2.10)

After setting (1),A=m ia & ' ' ,mi Eq. (2.10) can be written in compact matrix form as

'.TA

(2.11)

Eq. (2.6) and Eq. (2.7) yield the coefficient matrix of order xN as:

2 1 1

2 1

1 2 1

1 2 1

1 2

1 2 1

2

0

0

2

a a a

a a

a a a

a a a

a a

a a a

and in particular the column of the matrix ' read:




4 4

3 4 4

3

3

4

4

3 3 4

0 02 0

0 00

0 0

0 0' 1 , ' 2 , ' 3 , , ' 1 , ' .0 0

00

200 0

0 0 0

x x

a a

a a a

a

N Na

a

a

a a a

It is worth mentioning that tri-diagonal system (2.11) has unique solution as is invertible. We use

Thomas algorithm [47] to solve system (2.11) for weighting coefficients 1, ,

xi Na i . Similarly, the

coefficients 1

ib can be computed by considering the grid descritization in y direction.

One can determine these weighting coefficients using the second order spatial derivative approximation as

[36]: (2)

1

'' , , .x

N

mi i m Na i m

Similar to system (2.11), this system yields 2

, ,xi Na i .

Existence more than one basis to span a finite dimensional vector space, gives a chance to evaluate

2, ,

xi Na i with other base functions. Keeping computational cost in mind, we prefer Shu’s rth order

( 2r ) recursive formula based on polynomial DQM [40, 42] to compute

2 2, , ; , ,

x yi N i Na i b i as follows:

11 1

1,

11 1

1,

, , , ;

, , , .

, , , ;

, , , .

x

x

y

y

rr r i

i i ii N

i

Nr r

ii i N

i

rr r i

i i ii N

i

Nr r

ii i N

i

aa r a a i i

x x

a a i i

bb r b b i i

y y

b b i i

3 Implementation of the method to Burgers’ equation

3.1. 1D coupled viscous Burgers’ equation

On putting the values of derivatives approximate using MTB-DQM Eq. (1.1) can be re-written as:

(2) (1) (1) (1)

1 1 1 1

(2) (1) (1) (1)

1 1 1 1

, ( 0) ( ),

, ( 0) ( ), .

x x x x

ij ij ij ij

x x x x

ij ij ij ij x

N N N N

ij i j i j i j i i

j j j j

N N N N

ij i j i j i j i i N

j j j j

ua u u a u u a v v a u u t x

t

va v v a v u a v v a u v t x i

t

(3.12)

Keeping boundary conditions (1.2) in mind, Eq. (3.12) reduces to a set of first order ODEs:




1 1 1 1(2) (1) (1) (1)

2 2 2 2

1 1 1 1(2) (1) (1) (1)

2 2 2 2

; ( 0) ( ),

; (

x x x x

ij ij ij ij

x x x x

ij ij ij ij

N N N N

ij i j i j i j i i i

j j j j

N N N N

ij i j i j i j i i

j j j j

ua u u a u u a v v a u F u t x

t

va v v a v u a v v a u G v t

t

0) ( ), 1 .

i xx i N

(3.13)

Setting , , , ' , , 'i i i i i i i i i i i iu v u v u v , then

1 1 1 1

1 1 1 1

(2) (2) (1) (1) (1) (1) (1) (1)

1 1 1 1

(2) (2) (1) (1) (1) (1) (1) (1)

1 1 1 1

' ,

' .

i iN x i iN x i iN x i iN xx x x x

i iN x i iN x i iN x i iN xx x x x

i N i N i N i N

i N i N i N i N

F a u a u a u a u a v a v a u a u

G a v a v a v a v a v a v a u a u

3.2. 2D nonlinear coupled Burger’s equation

Similarly, on implementing MTB-DQM to Eq. (1.3)–(1.4) in space, we get

1 11 1(2) (2) (1) (1)

2 2 2 2

1 11 1(2) (2) (1) (1)

2 2 2 2

,

,

y yx x

ik jk ik jk

y yx x

ik jk ik jk

N NN Nij

kj ik ij kj ij ik i j

k k k k

N NN Nij

kj ik ij kj ij ik i j

k k k k

ua u b u u a u v b u F

t

va v b v u a v v b v G

t

( 0) ( , ), ( 0) ( , ), 1 , 1ij i j ij i j x yu t x x v t x x i N j N

(3.14)

where andij ij ij iju v and

1 1 1 1

1 1 1 1

(2) (2) (2) (2) (1) (1) (1) (1)

1 1 1 1

(2) (2) (2) (2) (1) (1) (1) (1)

1 1 1 1

,

.

i iN x j j N y i iN x j j N yx y x y

i iN x j j N y i iN x j j N yx y x y

i j j N j i iN ij j N j ij i iN

i j j N j i iN ij j N j ij i iN

F a u a u b u b u a u a u b u b u

G a v a v b v v u a v a v b v b v

(3.15)

The initial value systems of first order ODEs given in Eq. (3.13) or Eq. (3.14) can be solved by numerous

techniques, among others, we prefer SSP-RK54 scheme [48-49] to solve the system (3.13) and system

(3.14), through the following steps:

(1) 0.391752226571890 ( )m mu u tL u

(2) (1) (1)0.444370493651235 0.555629506348765 0.368410593050371 ( )mu u u tL u

(3) (2) (2)0.620101851488403 0.379898148511597 0.251891774271694 ( )mu u u tL u

(4) (3) (3) 0.178079954393132 0.821920045606868 0.544974750228521 ( )mu u u tL u

( 1) (2) (3) (3)

(4) (4)

0.517231671970585 0.096059710526147 0.063692468666290 ( )

0.386708617503269 0.226007483236906 ( )

mu u u tL u

u tL u

The SSP-RK54 scheme allows low storage and large domain of absolute properties [34-35, 52].

4 Stability analysis

The stability analysis for both dimension is being similar, we concerned with the stability analysis of 2D

coupled viscous Burgers’ equation (3.14), only. Let (2) (1)

2 1;ij ij

A a A a

&(2) (1)

2 1;ij ij

B b B b

be

matrices of weighting coefficients of order 2 2x yN N . Following [50], set i iu in the non

linear terms of Eq. (3.13), and andij ij ij iju v in the non linear terms of Eq. (3.14) and assumed to be




locally fixed. Then, Eq. (3.14) can be re-written as

.dU

U Hdt

(4.16)

Where

22 23 2( 1) 32 33 3( 1) ( 1)2 ( 1)3 ( 1)( 1)

22 23 2( 1) 32 33 3( 1) ( 1)2 ( 1)3 ( 1)( 1)

( , ) , ( , , , , , , , , , , , , )

( , , , , , , , , , , , , )

y y x x y

y y x x y

T

N N Nx N N N

N N Nx N N N

U u v u u u u u u u u u u

v v v v v v v v v v

,T

H F G , [ ], [ ], 1 ;1i j i j x yF F G G i N j N as defined in Eq. (3.15)

A O

BO A

, 1 1 2 2ij ijA A B A B , (4.17)

where 'O s null matrices, rA and rB ( 1,2)r are square block diagonal matrices (each of order

2 2x yN N ) of the weighting coefficients( ) ( ),r r

ij ija b respectively as given below:

1 1

1( ) ( )

1

1 1

[ ]; ; [ ],r r

r ij r ij

r

rr

r

C O O

O CA w I B C c

O

O O C

where I and 1 'O s are identity and null matrices of order ( 2)yN and ( 2)xN , respectively and

1 1

( ) ( )

2, ,i j x

r r

ij Nw a i j and

1 1

( ) ( )

2, , .i j y

r r

ij Nc b i j

It is worth mentioning that the stability of system (4.16) depends upon the eigenvalues of matrix [51].

The stability region is the set , ( ) 1,B

S z C R z z t where (.)R stability function and B

is the eigenvalue of . The stability region of SSP-RK54 scheme is depicted in Figure 1, see [Fig.5, 64].

The sufficient condition for the stability of system (4.16) is that B t belongs to the stability region of

SSP-RK54 scheme for each eigenvalue of .

Figure 1: Stability region of SSP-RK54 scheme with z t




Figure 2: Eigen values 1 2and for different grid size h

The eigenvalues of and 1,2r rA B r being identical nature, it is sufficient to compute the

eigenvalues i of ( 1,2)iA i . Figure 2 confirms that for different values of x yh h h , each eigenvalue

1 of matrix 1A is pure imaginary whereas each eigenvalue 2 of 2A is real and negative.

It is evident from Figure 2 and Eq. (4.17) that for a given values of &h , there exists t for which the

value t corresponding to each eigenvalue 2 1 0 02 of lies inside the stability region

of SSP-RK54. This shows that MTB-DQM produces stable solutions for the 2D coupled Burger’s

equation. Similarly, one can demonstrate that MTB-DQM produces stable solutions for the 1D coupled

Burger’s equation.

5 Numerical results and discussion

The accuracy and consistency of the scheme is measured in terms of* 2

2

1

: | |n

j j

j

L h U U

and

*L := max | |j jj

U U ( jU exact solution, *

jU computed solution at node jx ) error norms for

four test problems of (1+1)D and (2+1)D Burgers’ equation, which we compute by using C++.

Problem 1: On setting the parameters 1, 2 , Eq. (1.1) reduces to

2

2

2

2

2 , ( , ), t 0,

2 , ( , ), t 0,

uvu u uu x

t x x x

uvv v vv x

t x x x

(5.18)

The values of 1 2 3 4, , , , ,g g g g can be obtained from exact solution [7]:

-( , ) ( , ) sin( ) , [ , ], 0. tu x t v x t x e x t




Table 1: The comparison of order of convergence R and 2L and L of MTB-DQM for ( , )u x t = ( , )v x t of

Problem 1 for 1, 2 at 1.t

MTB-DQM

LBM [23]

xN 2L R L R 2L R L R

10 6.69E-03

3.27E-03

3.30E-02

1.15E-02

20 8.77E-04 2.93 4.40E-04 2.89 8.18E-03 2.0099 3.01E-03 1.9375

40 1.03E-04

3.08 5.13E-05

3.10 2.01E-03 2.0276 7.38E-04 2.0275

80 8.01E-06

3.68 4.15E-06

3.63 4.64E-04 2.1123 1.71E-04 2.1122

160 1.42E-06

2.50 7.02E-07

2.56 7.85E-05 2.5634 2.89E-05 2.5638

Table 2: Compression of 2L and L errors in Problem 1 for t = 0.001, N = 64.

T 2

L

L

MTB-DQM LBM [23] FDM [23] MTB-DQM LBM [23] FDM [23]

0.1 8.44E-06 3.03E-05 8.03E-05 2.16E-05 2.75E-05 7.27E-05

0.5 1.45E-05 1.52E-04 4.02E-04 1.62E-05 9.20E-05 2.44E-04

1.0 2.02E-05 3.03E-04 8.03E-04 1.02E-05 1.12E-04 2.95E-04

2.0 3.30E-05 6.07E-04 1.61E-03 4.20E-06 8.21E-05 2.18E-04

5.0 7.61E-05 1.52E-03 4.02E-03 4.87E-07 1.02E-05 2.71E-05

10.0 1.50E-04 3.04E-03 8.06E-03 6.64E-09 1.38E-07 3.66E-07

20.0 3.00E-04 6.09E-03 1.62E-02 6.10E-13 1.25E-11 3.34E-11

The computed solution is compared with LBM [23], order of convergence of the scheme is reported in

Table 1, which shows that order of convergence of MTB-DQM is cubic in space while LBM [23] is

quadratic. It is evident from Table 2 that the computed solutions are more accurate in comparison to FDM

[23] and LBM [23]. The comparison of 2L and L errors in MTB-DQM solutions with the errors from

existing schemes at different time levels 3,t with 0.001t andxN =121 , is reported in Table 3. It is

evident that produced solutions are more accurate than earlier schemes reported in [14, 16, 23-26].

Absolute error norms and solution behavior for this problem is depicted in Figure 3 for different time

levels.




Table 3: Comparison with earlier schemes in Problem 1 for u with 1, 2 at 1.t

Schemes

xN

t

0.5t 1t 2t 3t

L 2L L 2L L L

MTB-DQM 121 0.001 2.00E-07 7.82E-08 1.21E-07 8.45E-08 4.42E-08 1.62E-08

DQM[24] 0.01 1.52E-04 1.84E-04 1.35E-04 7.46E-05

CBC[16] 400 0.001 6.22E-06 1.02E-05 7.56E-06 2.04E-05

0.1t CBC[16] 200 0.001 4.10E-05 2.49E-05 8.21E-05 0.00003

FPM [14] 128 0.0001 1.16E-05 2.88E-05 L 2L

FFID[25] 200 0.001 1.79E-04 2.94E-04 2.17E-04 5.91E-04 5.30E-05 5.86E-05

CFDH[26] 16 0.001 7.27E-06 1.05E-08 2.38E-05 5.03E-07 3.77E-05 3.27E-08

MTB-DQM 121 0.001 2.00E-07 7.82E-08 1.21E-07 8.45E-08 3.01E-07 7.52E-08

Figure 3: Absolute errors and MTB-DQM solutions u v of Problem 1 at 3.0t for xN 121 ; 0.001t

Problem 2: Consider the 2D Burgers’ equations in 2

0,0.5 as in [27, 61] with

( , ) ,x y x y ( , )x y x y and boundary conditions can be extracted from the exact solution of this

problem [61]

2 2

2 2( , , ) , ( , , ) .

1 2 1 2

x y xt x y ytu x y t v x y t

t t

The MTB-DQM solutions are computed at 0.1t for40.025and 10x yh h t , and compared them

(u and v components) with exact solutions for 1/ 80Re in Tables 4 and 5 at different grid points

and time levels 0.1,0.3,0.5t . 2 ,L L errors for u , v are reported in Table 6. The findings show that the

computed results are agreed well with the exact solutions. The physical behavior of MTB-DQM solutions

of Problem 2 for u and v are depicted for Re 100 at 0.1t in Figure 4 and for Re 80 at 0.5t in

Figure 5.




Table 4: Comparison of exact & MTB-DQM solution u of Problem 2 with 4Re 80, 10 , 0.5.t t

Mesh 0.1t 0.3t 0.5t

MTB-DQM Exact MTB-DQM Exact MTB-DQM Exact

0.1,0.1 0.183673 0.183673 0.170732 0.170732 0.20004 0.20004

0.3,.01 0.346939 0.346939 0.268259 0.268259 0.19996 0.19996

0.2,0.2 0.367347 0.367347 0.341465 0.341465 0.40008 0.40008

0.4,0.2 0.530612 0.530612 0.438991 0.438991 0.40000 0.40000

0.5,0.3 0.714286 0.714286 0.609723 0.609723 0.60004 0.60004

0.3,0.4 0.653061 0.653061 0.634166 0.634166 0.80020 0.80020

0.4,0.5 0.836735 0.836735 0.804898 0.804898 1.00024 1.00024

0.5,0.5 0.918367 0.918367 0.853662 0.853662 1.00020 1.00020

Table 5: Comparison of exact & MTB-DQM solution v of Problem 2 with 4Re 80, 10 , 0.5.t t

Mesh

0.1t 0.3t 0.5t

MTB-DQM. Exact MTB-DQM Exact MTB-DQM Exact

0.1, 0.1 -0.02041 -0.02041 -0.07321 -0.07321 -0.20012 -0.20012

0.3, 0.1 0.183673 0.183673 0.170732 0.170732 0.20004 0.20004

0.2, 0.2 -0.04082 -0.04082 -0.14641 -0.14641 -0.40024 -0.40024

0.4, 0.2 0.163265 0.163265 0.097527 0.097527 -0.00008 -0.00008

0.5, 0.3 0.142857 0.142857 0.024321 0.024321 -0.2002 -0.2002

0.3, 0.4 -0.18367 -0.18367 -0.41479 -0.41479 -1.00056 -1.00056

0.4, 0.5 -0.20408 -0.20408 -0.48800 -0.48800 -1.20068 -1.20068

0.5, 0.5 -0.10204 -0.10204 -0.36603 -0.36603 -1.00060 -1.00060

Figure 4: The physical behavior of MTB-DQM solution of Problem 2 with Re 100 for ( , y, )u x t and ( , , )v x y t at

0.1t




Figure 5: MTB-DQM solution of Problem 2 with Re 80 for ( , y, )u x t and ( , , )v x y t at 0.05t

Table 6: 2L , L errors in ,u v of Problem 2 with Re 100 , various grid sizes and 410t at 0.01,0.5.t

For solution u For solution v

0.5t 0.01t 0.5t 0.01t

x(N , N )y

2L L 2L L

2L L 2L L

(4, 4) 6.23E-08 2.40E-08 1.48E-09 4.83E-10 6.23E-08 2.40E-08 1.48E-09 4.83E-10

(8, 8) 4.21E-09 1.21E-09 1.11E-10 2.27E-11

4.21E-09 1.21E-09 1.11E-10 2.27E-11

(17, 17) 2.66E-10 4.36E-11 7.55E-12 9.03E-13 2.66E-10 4.36E-11 7.55E-12 9.03E-13

(32, 32) 1.61E-11 1.40E-12 5.10E-13 3.13E-14

1.61E-11 1.40E-12 5.10E-13 3.13E-14

(44, 44) 4.13E-12 2.64E-13 1.60E-13 6.71E-15 4.13E-12 2.64E-13 1.60E-13 6.71E-15

(64, 64) 6.28E-13 2.28E-14 5.43E-14 1.80E-15

6.28E-13 2.28E-14 5.43E-14 1.80E-15

Problem 3: Consider 2D initial valued coupled viscous Burgers’ equations (1.3) in 2

0,0.5 with

( , ) sin( x) sin( ),x y y ( , )x y x y and together with the boundary conditions:

1 1 2 1(0, , ) cos( ), , , 1 cos( ); (0, , ) , , , , [0,0.5],

2 2 2

1 1 2 1( ,0, ) 1 sin( ); , , sin( ); ( ,0, ) ; , , , [0,0.5], 0.

2 2 2

yu y t y u y t y v y t y v y t y

xu x t x u x t x v x t x v x t x t

(5.19)

The numerical solutions are obtained at 0.1t for 40.025and 10x yh h t . In Table 7, MTB-DQM

solutionsu , v in Problem 3 with 50Re are compared with the solutions in [29, 30, 63] and the exact

solutions for grid size 20 20 and various time levels, and the computed solution behavior of u and v of

Problem 3 with 0.01 at 1,2,3t is depicted in Figure 6.




Figure 6: MTB-DQM solutions of Problem 3 for Re 100, 0.025,x yh h 4and 10 ,at 1,2,3t t

Table 7: Comparison for solutions ,u v in Problem 3 with 50Re , ( , ) (20,20)x xN N

4and 10 at 0.625.t t

Mesh

u

v

MTB-DQM FDM [30]

Expo-MCB-

DQM [63]

Jain-Holla

[29]

MTB-DQM FDM [30]

Expo-MCB-

DQM [63]

Jain-Holla

[29]

0.1,0.1 0.97056 0.97146 0.97056 0.97258 0.09842 0.09869 0.09842 0.09773

0.3,0.1 1.15152 1.15280 1.15152 1.16214 0.14107 0.14158 0.14107 0.14039

0.2,0.2 0.86244 0.86308 0.86243 0.86281 0.16732 0.16754 0.16732 0.16660

0.4,0.2 0.98078 0.97985 0.98078 0.96483 0.17223 0.17111 0.17223 0.17397

0.1,0.3 0.66336 0.66316 0.66335 0.66318 0.26380 0.26378 0.26380 0.26294

0.3,0.3 0.77226 0.77233 0.77226 0.77030 0.22653 0.22655 0.22653 0.22463

0.2,0.4 0.58273 0.58181 0.58273 0.58070 0.32935 0.32851 0.32935 0.32402

0.4,0.4 0.76179 0.75862 0.76179 0.74435 0.32884 0.32502 0.32884 0.31822

Problem 4: Consider 2D coupled viscous Burgers’ equation (1.3) in 2

0,1 with exact solution [6]

3 1 3 1, , ; , , ,

4 44 1 exp 4 4 Re/ 32 4 1 exp 4 4 Re/ 32

u x y t v x y t

x y t x y t

(5.20)

where and on , and , on can be computed from the exact solution.




Figure 7: MTB-DQM solution and exact solution u , v of Problem 4 with 0.01, 0.05h and 0.0001t at 1t

Figure 8: MTB-DQM solution and exact solution u , v of Problem 4 with 0.004, 0.05h and 0.0001t at 1t




Table 8: Comparison in errors, rate of convergence for vu, of Problem 4 with Re 100, 0.0001 t at 1.0t and different grid size

( , ).x xN N

solutionFor u

xN

mECDQ [45]

0.52

Exp-MCB-DQM [63]

10p MTB-DQM

mECDQ [45]

0.52

Exp-MCB- DQM[63]

10p MTB-DQM

2L R 2L R 2L R L R L R L R

4 1.0799E-02 1.5865E-02 1.6451E-02 2.1600E-03 2.3325E-03 2.8958E-03

8 2.6363E-03 2.03 1.8037E-03 3.13 1.9330E-03 3.08 2.8422E-04 2.93 1.6816E-04 3.79 1.9644E-04 3.89

16 3.5676E-04 2.89 3.8329E-04 2.23 3.9504E-04 2.29 3.6252E-05 3.44 1.9610E-05 3.10 2.0508E-05 3.26

32 6.2368E-05 2.52 8.0461E-05 2.52 8.1200E-05 2.28 2.4576E-06 3.42 2.1967E-06 3.15 2.2208E-06 3.21

64

1.0004E-05 2.64 1.5355E-05 2.38 1.5323E-05 2.41 2.1893E-07 3.49 2.1795E-07 3.33 2.1840E-07 3.35

solutionFor v

4 1.0799E-02 1.5865E-02 1.6451E-02 2.1600E-03 2.3325E-03 2.8958E-03

8 2.6363E-03 2.03 1.8037E-03 3.13 1.9330E-03 3.08 2.8422E-04 2.93 1.6816E-04 3.79 1.9644E-04 3.89

16 3.5676E-04 2.89 3.8329E-04 2.23 3.9504E-04 2.29 3.6252E-05 3.44 1.9610E-05 3.10 2.0508E-05 3.26

32 6.2368E-05 2.52 8.0461E-05 2.52 8.1200E-05 2.28 2.4576E-06 3.42 2.1967E-06 3.15 2.2208E-06 3.20

64

1.0004E-05 2.64 1.5355E-05 2.38 1.5323E-05 2.41 2.1893E-07 3.49 2.1795E-07 3.33 2.1840E-07 3.35

This problem is solved for 2 410 , 10 .t The computed 2L and L error norms are compared with

the errors due to mECDQ[45] and Expo-MCB-DQM [63] in Table 8, which shows that MTB-DQM

solutions are comparable to [45, 63], and are in good agreement with the exact solutions. MTB-DQM

solutions u and v and exact solutions of Problem 4 are depicted for 004.0,01.0 at 1t in Figure 7-

8.

6 Conclusions

In this paper, a new scheme: “modified trigonometric cubic B-spline DQM” has been developed for

numeric study of nonlinear PDEs. Specially, MTB-DQM has been implemented for both (1+1)D and

(2+1)D coupled Burgers’ equation. The weighting coefficients of the first-order derivative is obtained by

using MTB-DQM, and polynomial DQM has been adopted to evaluate the weighting coefficients for

second-order derivative, and so, MTB-DQM reduces the Burgers’ equation into set of first of order ODEs,

which we solve via SSP-RK54 scheme.

The findings shows that the computational cost of both MCB-DQM [31] and MTB-DQM is same, and the

computed solutions agreed well with the exact solutions as compared to the others schemes in 1D coupled

viscous Burgers equation and comparable to mECDQ [45] and Exp-MCB-DQM [63] for 2D coupled

viscous Burgers equation. The matrix stability analysis has also been carried out for various grid values,

which demonstrate that the proposed method is stable for coupled viscous Burgers equation. It is worth

mentioning that MTB-DQM solutions of Burgers’ equation are computed without any linearization

technique and transformation.




Acknowledgement

The authors are grateful to the anonymous referees for their time, effort, and useful technical comments

and valuable suggestions to improve readability of the paper, which led to a significant improvement of the

paper .P. Kumar is also thankful to Babasaheb Bhimrao Ambedkar University Lucknow India, for

financial support to carry out the work.

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an algorithm based on dqm with modified trigonometric cubic b-

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