Error Estimates to Smooth Solutions of Runge–Kutta Discontinuous Galerkin Method for Symmetrizable Systems of Conservation Laws

2006 ◽  
Vol 44 (4) ◽  
pp. 1703-1720 ◽  
Author(s):  
Qiang Zhang ◽  
Chi‐Wang Shu
Keyword(s):  
Conservation Laws ◽  
Galerkin Method ◽  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Smooth Solutions ◽  
Systems Of Conservation Laws ◽  
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.

scholarly journals A priori error estimates of Adams-Bashforth discontinuous Galerkin Methods for scalar nonlinear conservation laws

2018 ◽  
Vol 26 (3) ◽  
pp. 151-172
Author(s):  
Charles Puelz ◽  
Béatrice Rivière
Keyword(s):  
Conservation Laws ◽  
Galerkin Method ◽  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Nonlinear Conservation Laws ◽  
Priori Error Estimates ◽  
Forward Euler

Abstract In this paper we show theoretical convergence of a second-order Adams-Bashforth discontinuous Galerkin method for approximating smooth solutions to scalar nonlinear conservation laws with E-fluxes. A priori error estimates are also derived for a first-order forward Euler discontinuous Galerkin method. Rates are optimal in time and suboptimal in space; they are valid under a CFL condition.

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