An h-adaptive RKDG method with troubled-cell indicator for one-dimensional detonation wave simulations

In this work, we discuss an extension of the adaptive technique in Zhu and Qiu (J. Comput. Phys. 228, 6957-6976 2009) to design an h-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method for the simulations of several classical one-dimensional detonation waves. The TVB troubled-cell indicator is employed to detect the troubled cells which are believed to contain the discontinuities. An adaptive mesh is generated at each time-level by refining the troubled cells and coarsening the others. A recursive multi-level mesh refinement technique is designed to avoid the problem that the detonation front moves so fast that there are not enough cells to resolve the detonation front before it leaves. We describe the numerical implementation in detail including the adaptive procedure, solution reconstruction method and troubled-cell indicator. Furthermore, a high order positivity-preserving technique is employed for the robustness of our algorithm. Extensive numerical tests are conducted to show the effectiveness of the adaptive strategy and advantages of our adaptive method over the fixed-mesh RKDG method in saving the computational storage and improving the solution quality.