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Description: TITLE: ON THE DEVELOPMENT OF HIGHER ORDER STRONG STABILITY PRESERVING TIME EVOLUTION METHOD OF PDE

Abstract: Strong-Stability-Preserving Runge-Kutta methods (SSP-RK) are popular time-stepping
schemes that are widely used for evolving the numerical solution of hyperbolic conservation
law, in particular where solutions have discontinuities or sharp gradients.

Compared to standard RK methods, the methods (under some conditions) offer
advantages such as maintaining nonlinear stability with respect to the total variation and
preserving monotonicity conditions with respect to some convex functional.

With nonnegative coefficients, explicit SSP-RK methods are known to exist only
up to fourth order accurate, where implicit versions are only up to sixth order accuracy. In order
to break these SSP accuracy barriers, we consider allowing negative coefficients (which requires a modified formulation of the spatial discretization) and use numerical optimization to find optimal methods. We extend the SSP theory to additive problem and construct Implicit-Explicit (IMEX) methods. Based on sufficient conditions for an additive method to be SSP, we formulate an optimization problem to obtain optimal SSP IMEX methods. Finally, we construct efficient embedded pairs for optimal Explicit, Implicit, and IMEX SSP methods to be used for error control and the methods will be tested on a variety of scalar hyperbolic partial differential equations.