Wave reflections due to hp-nonuniform mesh transitions

Adaptive grids can reduce the prohibitive cost of high-fidelity turbulence simulations, but these nonuniform grids can lead to issues, particularly when combined with high-order methods. We focus on the Discontinuous Galerkin Method and study one issue that arises when solving 2D linear hyperbolic systems on nonuniform grids. We analyze a plane wave incident on a grid interface and find that physical waves in families other than that of the incident wave can be erroneously produced. In contrast to the 1D case, no choice of numerical flux prevents these errors, and erroneous waves can be in physical wave modes that travel away from the grid interface. Analyzing the acoustic limit of the 2D Euler equations, we find that an incident acoustic wave causes an erroneous reflected acoustic wave. In the context of a fine-coarse interface, the reflected wave's amplitude can be significant relative to the fine grid's error tolerance for p≧2. In the context of an acoustic wave transmitted from a given grid to a finer grid, errors in the amplitude and phase of the transmitted wave can be significant relative to the original grid's error tolerance. We find that both reflection and transmission errors are greater for higher p, because the effective change in resolution across a given mesh size ratio is greater. We analyze h-, p-, and hp-nonuniform grids and find that combined interfaces in h and p can decrease errors relative to interfaces in h alone. In the more general case of the 2D Euler equation, we find that an incident acoustic wave can produce an erroneous reflected or transmitted shear wave, and vice versa. However, entropy waves produce no erroneous reflected or transmitted waves, as long as a characteristics-splitting numerical flux is used. We find no solution to exactly mitigate these errors due to grid interfaces, so we recommend considering them when determining resolution requirements for high-order adaptive simulations.