Volume 24, Issue 2
High Resolution Schemes for Conservation Laws and Convection-Diffusion Equations with Varying Time and Space Grids
DOI:

J. Comp. Math., 24 (2006), pp. 121-140.

Published online: 2006-04

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• Abstract

This paper presents a class of high resolution local time step schemes for nonlinear hyperbolic conservation laws and the closely related convection--diffusion equations, by projecting the solution increments of the underlying partial differential equations (PDE) at each local time step. The main advantages are that they are of good consistency, and it is convenient to implement them. The schemes are $L^\infty$ stable, satisfy a cell entropy inequality, and may be extended to the initial boundary value problem of general unsteady PDEs with higher--order spatial derivatives. The high resolution schemes are given by combining the reconstruction technique with a second order TVD Runge-Kutta scheme or a Lax-Wendroff type method, respectively. The schemes are used to solve a linear convection--diffusion equation, the nonlinear in-viscid Burgers' equation, the one-- and two--dimensional compressible Euler equations, and the two--dimensional incompressible Navier--Stokes equations. The numerical results show that the schemes are of higher--order accuracy, and efficient in saving computational cost, especially, for the case of combining the present schemes with the adaptive mesh method \cite{TANGT}. The correct locations of the slow moving or stronger discontinuities are also obtained, although the schemes are slightly nonconservative.

• Keywords

Hyperbolic conservation laws Degenerate diffusion High resolution scheme Finite volume method Local time discretization

@Article{JCM-24-121, author = {}, title = {High Resolution Schemes for Conservation Laws and Convection-Diffusion Equations with Varying Time and Space Grids}, journal = {Journal of Computational Mathematics}, year = {2006}, volume = {24}, number = {2}, pages = {121--140}, abstract = { This paper presents a class of high resolution local time step schemes for nonlinear hyperbolic conservation laws and the closely related convection--diffusion equations, by projecting the solution increments of the underlying partial differential equations (PDE) at each local time step. The main advantages are that they are of good consistency, and it is convenient to implement them. The schemes are $L^\infty$ stable, satisfy a cell entropy inequality, and may be extended to the initial boundary value problem of general unsteady PDEs with higher--order spatial derivatives. The high resolution schemes are given by combining the reconstruction technique with a second order TVD Runge-Kutta scheme or a Lax-Wendroff type method, respectively. The schemes are used to solve a linear convection--diffusion equation, the nonlinear in-viscid Burgers' equation, the one-- and two--dimensional compressible Euler equations, and the two--dimensional incompressible Navier--Stokes equations. The numerical results show that the schemes are of higher--order accuracy, and efficient in saving computational cost, especially, for the case of combining the present schemes with the adaptive mesh method \cite{TANGT}. The correct locations of the slow moving or stronger discontinuities are also obtained, although the schemes are slightly nonconservative. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8739.html} }
TY - JOUR T1 - High Resolution Schemes for Conservation Laws and Convection-Diffusion Equations with Varying Time and Space Grids JO - Journal of Computational Mathematics VL - 2 SP - 121 EP - 140 PY - 2006 DA - 2006/04 SN - 24 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8739.html KW - Hyperbolic conservation laws KW - Degenerate diffusion KW - High resolution scheme KW - Finite volume method KW - Local time discretization AB - This paper presents a class of high resolution local time step schemes for nonlinear hyperbolic conservation laws and the closely related convection--diffusion equations, by projecting the solution increments of the underlying partial differential equations (PDE) at each local time step. The main advantages are that they are of good consistency, and it is convenient to implement them. The schemes are $L^\infty$ stable, satisfy a cell entropy inequality, and may be extended to the initial boundary value problem of general unsteady PDEs with higher--order spatial derivatives. The high resolution schemes are given by combining the reconstruction technique with a second order TVD Runge-Kutta scheme or a Lax-Wendroff type method, respectively. The schemes are used to solve a linear convection--diffusion equation, the nonlinear in-viscid Burgers' equation, the one-- and two--dimensional compressible Euler equations, and the two--dimensional incompressible Navier--Stokes equations. The numerical results show that the schemes are of higher--order accuracy, and efficient in saving computational cost, especially, for the case of combining the present schemes with the adaptive mesh method \cite{TANGT}. The correct locations of the slow moving or stronger discontinuities are also obtained, although the schemes are slightly nonconservative.