2014/03/25 by Sigal Gottlieb, Gottlieb, Sigal, Zachary J. Grant +3
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1403.6519
openalex publication_date 2014/03/25 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
High order spatial discretizations with monotonicity properties are often\ndesirable for the solution of hyperbolic PDEs. These methods can advantageously\nbe coupled with high order strong stability preserving time discretizations.\nThe search for high order strong stability time-stepping methods with large\nallowable strong stability coefficient has been an active area of research over\nthe last two decades. This research has shown that explicit SSP Runge--Kutta\nmethods exist only up to fourth order. However, if we restrict ourselves to\nsolving only linear autonomous problems, the order conditions simplify and this\norder barrier is lifted: explicit SSP Runge--Kutta methods of any linear order\nexist. These methods reduce to second order when applied to nonlinear problems.\nIn the current work we aim to find explicit SSP Runge--Kutta methods with large\nallowable time-step, that feature high linear order and simultaneously have the\noptimal fourth order nonlinear order. These methods have strong stability\ncoefficients that approach those of the linear methods as the number of stages\nand the linear order is increased. This work shows that when a high linear\norder method is desired, it may be still be worthwhile to use methods with\nhigher nonlinear order.\n