Volume 6, Issue 3 (Vol. 6, No. 3 2020)
mmr 2020, 6(3): 449-464 |
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1- Isfahan University of Technology , mokhtari@cc.iut.ac.ir
2- Isfahan University of Technology
2- Isfahan University of Technology
Abstract: (2038 Views)
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:
We encapsulate findings and conclusions of this research as follows:
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.
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.
Type of Study: Research Paper |
Subject:
alg
Received: 2017/12/2 | Revised: 2020/12/14 | Accepted: 2019/05/8 | Published: 2020/11/30 | ePublished: 2020/11/30
Received: 2017/12/2 | Revised: 2020/12/14 | Accepted: 2019/05/8 | Published: 2020/11/30 | ePublished: 2020/11/30
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