A parallel p-adaptive discontinuous Galerkin method for the Euler equations with dynamic load-balancing 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 $\left(p\right)$ and $\left(p-1\right)$ is used to adapt degrees of freedom in each computational element, which remarkably reduces the computational cost while still maintaining an accurate solution. The developed method is implemented with under the Charm++ parallel computing framework. Charm++ is a parallel computing framework that includes various load-balancing strategies. Implementing the numerical solver under Charm++ system provides us with access to a suite of dynamic load balancing strategies. This 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. It is observed that the unbalanced load distribution caused by the parallel p-adaptive DG method can be alleviated by the dynamic load balancing from Charm++ system. Due to this, high performance gain can be achieved. For the testcases studied in the current work, the parallel performance gain ranged from 1.5× to 3.7×. Therefore, the developed p-adaptive DG method can significantly reduce the total simulation time in comparison to the standard DG method without p-adaptation.