Efficient High-Order Methods for Solution of Hyperbolic Problems

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Physical processes around us, for example the movement of weather fronts, can be described using mathematical objects called partial differential equations. These equations are also used by engineers in developing new products or improving existing ones; for example they can be used to calculate lift and drag on an airplane wing. Such equations are too complex to be solved exactly, so we employ numerical methods to solve them approximately. ***The aim of this proposal is the development, analysis and implementation of the numerical method called the discontinuous Galerkin (DG) method. We propose to work on a better mathematical understanding of the DG method when applied to hyperbolic conservation laws and to make it more efficient and robust. The DG method is highly accurate in right conditions but might require more computational resources than other methods. We seek to understand why exactly it is so much more accurate by analyzing its dispersive and dissipative properties on unstructured computational meshes in two- and three-dimensional spaces. With this understanding, we will derive new methods that might have a better trade-off in terms of cost and accuracy, especially when applied to problems that develop shock waves.***We will also work on components of the scheme such as new stabilization techniques, as well as error estimation and adaptive techniques. These are needed to make the method more efficient and more suitable for practical applications. Estimation of the error that the scheme commits is needed in order to produce an accurate solution and to distribute computational resources in the most advantageous way. Stabilization techniques are used to suppress non-physical oscillations in the numerical solutions near discontinuities. Absence of a robust method of scheme stabilization is one of the main issues to making preventing the scheme from attaining its full potential.***The expected result of this proposal is the advancement of the field of numerical analysis. We also expect that the developed algorithms and approaches will lead to faster and more efficient computations with applications to compressible flow problems.