Path Conservative WENO Schemes and Riemann Solvers for Continuum Mechanics

2019 ◽  
Author(s):  
Ariunaa Uuriintsetseg ◽  
Michael Dumbster
Keyword(s):  
Continuum Mechanics ◽  
Weno Schemes ◽  
Riemann Solvers

scholarly journals Complete Riemann Solvers for the Hyperbolic GPR Model of Continuum Mechanics

2021 ◽  
Vol 35 ◽  
pp. 60-72
Author(s):  
U. Ariunaa ◽  
◽  
M. Dumbser ◽  
Ts. Sarantuya ◽  
◽  
...  
Keyword(s):  
Continuum Mechanics ◽  
Riemann Solver ◽  
Governing Equations ◽  
Overdetermined System ◽  
Riemann Solvers ◽  
First Order ◽  
Conservative Schemes ◽  
Solid Dynamics ◽  
First Time ◽  
Better Than

In this paper, complete Riemann solver of Osher-Solomon and the HLLEM Riemann solver for unified first order hyperbolic formulation of continuum mechanics, which describes both of fluid and solid dynamics, are presented. This is the first time that these types of Riemann solvers are applied to such a complex system of governing equations as the GPR model of continuum mechanics. The first order hyperbolic formulation of continuum mechanics recently proposed by Godunov S. K., Peshkov I. M. and Romenski E. I., further denoted as GPR model includes a hyperbolic formulation of heat conduction and an overdetermined system of PDE. Path-conservative schemes are essential in order to give a sense to the non-conservative terms in the weak solution framework since governing PDE system contains non-conservative products, too. New Riemann solvers are implemented and tested successfully, which means it certainly acts better than standard local Lax-Friedrichs-type or Rusanov-type Riemann solvers. Two simple computational examples are presented, but the obtained computational results clearly show that the complete Riemann solvers are less dissipative than the simple Rusanov method that was employed in previous work on the GPR model.

A First Course in Continuum Mechanics

2001 ◽  
Author(s):  
Oscar Gonzalez ◽  
Andrew M. Stuart
Keyword(s):  
Continuum Mechanics

scholarly journals Time-varying Riemann solvers for conservation laws on networks

2009 ◽  
Vol 247 (2) ◽  
pp. 447-464 ◽  
Author(s):  
Mauro Garavello ◽  
Benedetto Piccoli
Keyword(s):  
Conservation Laws ◽  
Time Varying ◽  
Riemann Solvers

Essentially non-oscillatory and weighted essentially non-oscillatory schemes

Acta Numerica ◽  
2020 ◽  
Vol 29 ◽  
pp. 701-762
Author(s):  
Chi-Wang Shu
Keyword(s):  
Main Idea ◽  
High Order Accuracy ◽  
Finite Difference Schemes ◽  
Approximation Procedure ◽  
Discontinuous Solutions ◽  
Weno Schemes ◽  
Convection Diffusion ◽  
Different Types ◽  
Basic Ideas ◽  
Weighted Eno

Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities sharply and in an essentially non-oscillatory fashion. Both finite volume and finite difference schemes have been designed using the ENO or WENO procedure, and these schemes are very popular in applications, most noticeably in computational fluid dynamics but also in other areas of computational physics and engineering. Since the main idea of the ENO and WENO schemes is an approximation procedure not directly related to partial differential equations (PDEs), ENO and WENO schemes also have non-PDE applications. In this paper we will survey the basic ideas behind ENO and WENO schemes, discuss their properties, and present examples of their applications to different types of PDEs as well as to non-PDE problems.

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