High order entropy preserving ADER-DG schemes. (English) Zbl 1511.65109
Summary: In this paper, we develop a fully discrete entropy preserving ADER-Discontinuous Galerkin (ADER-DG) method. To obtain this desired result, we equip the space part of the method with entropy correction terms that balance the entropy production in space, inspired by the work of Abgrall. Whereas for the time-discretization we apply the relaxation approach introduced by Ketcheson that allows to modify the timestep to preserve the entropy to machine precision. Up to our knowledge, it is the first time that a provable fully discrete entropy preserving ADER-DG scheme is constructed. We verify our theoretical results with various numerical simulations.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 35L65 | Hyperbolic conservation laws |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
Keywords:
hyperbolic conservation laws; ADER; discontinuous Galerkin; entropy conservation/stability; structure preserving; relaxation methodSoftware:
RRK_rrReferences:
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