Discovery and optimization of low-storage Runge-Kutta methods
Fletcher, Matthew T.
Wilcox, Lucas C.
Kozdon, Jeremy E.
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Runge-Kutta (RK) methods are an important family of iterative methods for approximating the solutions of ordinary differential equations (ODEs) and differential algebraic equations (DAEs). It is common to use an RK method to discretize in time when solving time dependent partial differential equations (PDEs) with a method-of-lines approach as in Maxwell’s Equations. Different types of PDEs are discretized in such a way that could result in a high dimensional ODE or DAE.We use a low-storage RK (LSRK) method that stores two registers per ODE dimension, which limits the impact of increased storage requirements. Classical RK methods, however, have one storage variable per stage. In this thesis we compare the efficiency and accuracy of LSRK methods to RK methods. We then focus on optimizing the truncation error coefficients for LSRK to discover new methods. Reusing the tools from the optimization method, we discover new methods for low-storage half-explicit RK (LSHERK) methods for solving DAEs.
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