High Order Maximum Principle Preserving Finite Difference Schemes for Hyperbolic Conservation Laws

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The main focus of this proposal is to develop and to analyze a novel parametrized maximum principle preserving flux limiter technique for high order numerical schemes applied to hyperbolic conservation laws. A flux limiting technique will also be designed to obtain high order positivity preserving schemes. Numerical schemes that preserve the maximum principle and positivity are desirable because physically relevant solutions have those properties. The development is based on finite difference methods, which have the advantage of producing accurate approximations with low computational cost especially in multi-dimensional simulations. Within the proposed framework, conservative maximum principle preserving high order finite difference, finite volume and discontinuous Galerkin schemes can be designed that allow for significantly large CFL number, and therefore more efficient computational simulation. Some important applications investigated in this proposal include compressible Euler equations, magneto hydrodynamics equations and Vlasov-Maxwell equations.The investigator is developing new computational techniques that can be applied to difficult and very important problems in science and engineering. These techniques address shortcomings in existing methods and should allow more efficient, robust, and accurate computer simulations in a number of critical applications. One such application is the supersonic flow problem, which is of great importance in designing astrophysical jets and also in the simulation of reentry vehicle for space flight modeling. Another application is the study of the magneto hydrodynamic systems, which arises in space weather modeling, in modeling electric propulsion sources, and in systems involving plasma (such as plasma-opening switches, flight control via plasma, plasma assisted combustion). These problems are strategically important for the design of the next generation devices of great industrial and commercial value.