A Riemann-solver-free Runge-Kutta Discontinuous Galerkin Method for Conservation Laws

2017 ◽  
Author(s):  
Shuangzhang Tu
Keyword(s):  
Conservation Laws ◽  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Riemann Solver ◽  
Runge Kutta

Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws

2019 ◽  
Vol 53 (1) ◽  
pp. 105-144 ◽  
Author(s):  
Lingling Zhou ◽  
Yinhua Xia ◽  
Chi-Wang Shu
Keyword(s):  
Conservation Laws ◽  
Galerkin Method ◽  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Second Order ◽  
Arbitrary Lagrangian Eulerian ◽  
Third Order ◽  
Fully Discrete ◽  
Runge Kutta

In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh2 and the second order TVD-RK scheme needs $ \tau \le \rho {h}^{\frac{4}{3}}$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h.

Runge-Kutta Discontinuous Galerkin Method with Front Tracking Method for Solving the Compressible Two-Medium Flow on Unstructured Meshes

2016 ◽  
Vol 9 (1) ◽  
pp. 73-91 ◽  
Author(s):  
Haitian Lu ◽  
Jun Zhu ◽  
Chunwu Wang ◽  
Ning Zhao
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Front Tracking ◽  
Unstructured Meshes ◽  
Interfacial Region ◽  
Tracking Method ◽  
Front Tracking Method ◽  
Medium Flow ◽  
Runge Kutta

AbstractIn this paper, we extend using the Runge-Kutta discontinuous Galerkin method together with the front tracking method to simulate the compressible two-medium flow on unstructured meshes. A Riemann problem is constructed in the normal direction in the material interfacial region, with the goal of obtaining a compact, robust and efficient procedure to track the explicit sharp interface precisely. Extensive numerical tests including the gas-gas and gas-liquid flows are provided to show the proposed methodologies possess the capability of enhancing the resolutions nearby the discontinuities inside of the single medium flow and the interfacial vicinities of the two-medium flow in many occasions.

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