Solving a System of 2D Burger's Equations using Semi-Lagrangian Finite Difference Schemes

Authors
Reza Mokhtari
Elham Feizollahi
Isfahan University of Technology
10.52547/mmr.6.3.449
Abstract
Introduction

Following and generalizing the excellent work of Wang et ‎al. ‎[26], ‎we extract here some new scheme‎s, ‎based on the‎ ‎semi-Lagrangian discretization‎, ‎the modified equation theory‎, ‎and‎ ‎the local one-dimensional (LOD) scheme for computing solutions to a‎ ‎system of two-dimensional (2D) Burgers' equations‎. ‎A careful error‎ ‎analysis is carried out to demonstrate the accuracy of the‎ ‎proposed semi-Lagrangian finite difference methods‎. ‎By conducting‎ ‎numerical simulation to the nonlinear system of 2D Burgers‎’ ‎equations (3.1), ‎we show high accuracy and‎ ‎unconditional stability of the five-point implicit scheme (3.32-3.33)‎. ‎The results of‎ [26] and this paper confirm that the classical modified‎ ‎equation technique can be easily extended to various 1D as well as 2D‎ ‎nonlinear problems‎. ‎Furthermore, a new viewpoint is opened to‎ ‎develop efficient semi-Lagrangian methods‎. ‎Without using suitable‎ ‎interpolants for generating the solution values at the departure‎ ‎points‎, ‎we are not able to apply our method‎. ‎Instead of focusing our‎ ‎concentration on dealing with the effect of various interpolation‎ ‎methods‎, ‎we focus our attention on constructing some‎ ‎explicit and implicit schemes‎. ‎Among various interpolants which‎ ‎can be found in the literature [6], [21], ‎we just‎ ‎exploit the simplest and more applicable interpolants‎, ‎i.e.‎, ‎B-spline and Lagrange interpolants‎. Some semi-Lagrangian schemes are developed using the modified equation‎ ‎approach‎, i.e., ‎a six-point explicit method (which suffers from the‎ ‎limited stability condition)‎, ‎a six-point implicit method (which‎ ‎has unconditional stability but low order truncation error)‎, ‎and a‎ ‎five-point implicit method‎ (3.32-3.33) which has‎ ‎unconditional stability and high order truncation error‎. ‎In each‎ ‎step of this scheme, we must solve two tridiagonal linear systems‎ ‎and therefore its computational complexity is low‎. ‎Furthermore, it‎ ‎can be implemented in parallel‎. ‎As mentioned in [26], ‎this algorithm can be naturally‎ ‎extended to the development of efficient and accurate‎ ‎semi-Lagrangian schemes for many other types of nonlinear‎ ‎time-dependent problems‎, ‎such as the KdV equation and ‎Navier-Stokes equations‎, ‎where advection plays an important role‎. ‎We tried in [9] to apply this approach to the KdV equation but‎ ‎constructing an implicit method which has unconditional stability‎ ‎and high order truncation error needs some considerable symbolic‎ ‎computations for extracting the coefficients of the scheme‎.

Material and methods

For constructing five-point implicit scheme‎ (3.32-3.33), we need to exploit Lagrange or B-spline interpolation method, ‎‎modified equation approach‎ and ‎local‎ ‎one-dimensional technique. The five-point implicit scheme is unconditional stable, has satisfactory order of convergence and its computational costs is low.

Results and discussion

Using the modified equation‎ ‎approach, some semi-Lagrangian schemes for solving a‎ ‎system of 2D Burgers' equations are developed here which are:

A six-point explicit method which is conditionally stable ‎ and its order of truncation error is low,
‎A six-point implicit method which‎ ‎has unconditional stability and its order of truncation error is not high‎,
A five-point implicit method‎ which has‎ ‎unconditional stability, high order truncation error and resonable computational complexity‎.

Conclusion

We encapsulate findings and conclusions of this research as follows:

Our‎ ‎proposed scheme is a local one-dimensional scheme which‎ ‎obtained on the basis of the modified equation approach,
Our semi-Lagrangian finite‎ ‎difference scheme is not limited by the‎ ‎Courant‎- ‎Friedrichs-Lewy (CFL) condition and therefore we can‎ ‎apply larger step size for the time variable,
The five-point implicit method‎ proposed is a‎ high order ‎unconditionally stable method with resonable computational costs‎.
Keywords
System of 2D Burgers' equations‎
‎Semi-Lagrangian‎ ‎finite difference scheme‎
‎Modified equation approach‎
Local‎ ‎one-dimensional approach‎‎
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Mathematical Research
Volume 6, Issue 3 - Serial Number 14
Autumn 2020
Pages 449-464