A Dynamically Load-balanced Parallel p-adaptive Discontinuous Galerkin Method for the Compressible Euler Equations on Tetrahedral Grids

A novel p-adaptive discontinuous Galerkin (DG) method has been developed to solve the Euler equations on three-dimensional tetrahedral grids. Hierarchical orthogonal basis functions are adopted for the DG spatial discretization while a third order TVD Runge-Kutta method is used for the time integration. A vertex-based limiter is applied to the numerical solution in order to eliminate oscillations in the high order method. An error indicator constructed from the solution of order (\mathbit{p}) and (\mathbit{p}-\mathbf{1}) is used to adapt degrees of freedom in each computational element, which remarkably reduces the computational cost while still maintaining an accurate solution. In terms of the parallel implementation, the developed method is implemented with the Charm++ parallel computing library. The Charm++ runtime system allows dynamic load-balancing automatically, which can be efficiently used to alleviate the load imbalances created by p-adaptation. A number of numerical experiments are performed to demonstrate both the numerical accuracy and parallel performance of the developed p-adaptive DG method. The numerical results show that the developed \mathbit{p}-adaptive method is capable of achieving high performance gains with the help of the automatic dynamic load balancing strategies available from the Charm++ library. Therefore, the developed p-adaptive DG method can significantly reduce the total simulation time in comparison to the standard DG method without p-adaptation.