scholarly journals Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Analysis and application in one dimension

2016 ◽  
Vol 86 (305) ◽  
pp. 1203-1232 ◽  
Author(s):  
Christian Klingenberg ◽  
Gero Schnücke ◽  
Yinhua Xia
Keyword(s):  
Conservation Laws ◽  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
One Dimension ◽  
Arbitrary Lagrangian Eulerian

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.

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