Robust and Efficient High Order Methods for Time Dependent Problems

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Robust and efficient high order accurate methods for computation have gained more and more popularity in the numerical modeling of real world problems for their ability to produce high fidelity simulations. However, such methods are not available or not well understood for hydrodynamics equations modeling high speed flows in spacecraft design, combustion, detonation, astrophysical jets, hurricanes, tsunamis, plasma dynamics, and inertial confinement fusion. This research project aims to develop improved numerical methods for simulation of these systems. Progress in designing robust and more efficient high order methods will significantly impact on simulation technology for such applications. The state of high order accurate numerical methods is still far from being practically satisfactory for time-dependent nonlinear problems. Compared to their low order counterparts, high order methods are much harder to stabilize and might be less efficient in practice due to much larger computer memory cost. Thus it remains challenging to utilize a high order accurate method to solve nonlinear hydrodynamics equations in real world problems. The objective of this proposal is to address these real-world problem challenges from specific perspectives. First of all, one would like to ensure the robustness of Eulerian schemes by preserving certain invariances of physical quantities such as positivity. Second, one would like to design more efficient implementations of very high order schemes on curved elements for complex geometries.