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2009
Abstract Strong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. SSP methods preserve the strong stability properties—in any norm, seminorm or convex functional—of the spatial discretization coupled with first order Euler time stepping.
2009 •
The aim of this thesis is the development of high order numerical methods 1 for hyperbolic PDEs. One of the principal difficulties in solving hyperbolic PDEs is the handling of discontinuities, which tend to lead to spurious oscillations and numerical instability. This thesis is largely concerned with methods developed to avoid such oscillations. In this chapter we give background information and motivation for the numerical methods developed in this thesis. We also provide an outline of the remainder of the thesis.
2005 •
Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties–in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping.
Journal of Scientific Computing
A Numerical Study of Diagonally Split Runge-Kutta Methods for PDEs with Discontinuities2008 •
Diagonally split Runge–Kutta (DSRK) time discretization methods are a class of implicit time-stepping schemes which offer both high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge–Kutta or linear multistep methods and therefore appears promising for the strong stability preserving (SSP) time-stepping community which is generally concerned with computing oscillation-free numerical solutions of PDEs. Using a variety of numerical test problems, we show that although second- and third-order unconditionally contractive DSRK methods do preserve the strong stability property for all time step-sizes, they suffer from order reduction at large step-sizes. Indeed, for time-steps larger than those typically chosen for explicit methods, these DSRK methods behave like first-order implicit methods. This is unfortunate, because it is precisely to allow a large time-step that we choose to use implicit methods. These results suggest that unconditionally contractive DSRK methods are limited in usefulness as they are unable to compete with either the first-order backward Euler method for large step-sizes or with Crank–Nicolson or high-order explicit SSP Runge–Kutta methods for smaller step-sizes. We also present stage order conditions for DSRK methods and show that the observed order reduction is associated with the necessarily low stage order of the unconditionally contractive DSRK methods.
2008 •
Strong stability-preserving (SSP) Runge-Kutta methods were developed for time integration of semidiscretizations of partial differential equations. SSP methods preserve stability properties satisfied by forward Euler time integration, under a modified time-step restriction. We consider the problem of finding explicit Runge-Kutta methods with optimal SSP time-step restrictions, first for the case of linear autonomous ordinary differential equations and then for nonlinear or nonautonomous equations.
Applied Numerical Mathematics
Strong-stability-preserving 3-stage Hermite–Birkhoff time-discretization methods2011 •
2008 •
2012 •
Abstract: We apply the concept of effective order to strong stability preserving (SSP) explicit Runge--Kutta methods. Relative to classical Runge--Kutta methods, effective order methods are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. The relaxed order conditions allow for greater freedom in the design of effective order methods.
Mathematics of Computation
On stability issues for IMEX schemes applied to 1D scalar hyperbolic equations with stiff reaction terms2011 •
Journal of Scientific Computing
Multirate Explicit Adams Methods for Time Integration of Conservation Laws2009 •
2010 •
The method of lines approach for solving hyperbolic conservation laws is based on the idea of splitting the discretization process in two stages. First, the spatial discretization is performed by leaving the system continuous in time. This approximation is usually developed in a non-oscillatory manner with a satisfactory spatial accuracy. The obtained semi-discrete system of ordinary differential equations (ODE) is then solved by using some standard time integration method. However, not all of them give satisfactory results. In the last few years, a series of papers appeared, dealing with the high order strong stability preserving (SSP) time integration methods that maintain the total variation diminishing (TVD) property of the first order forward Euler method. In this work the optimal SSP Runge{; ; ; Kutta methods of di®erent order are considered in combination with the ¯nite volume weighted essentially non-oscillatory (WENO) discretization. Furthermore, a new semi{; ; ; implicit W...
2006 •
2007 •
2018 •
Lecture Notes in Computer Science
Explicit Time Stepping Methods with High Stage Order and Monotonicity Properties2009 •
Journal of Scientific Computing
Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order2017 •
Journal of Scientific Computing
Multirate Timestepping Methods for Hyperbolic Conservation Laws2007 •
2013 •
2014 •
2001 •
Journal of Scientific Computing
High-Order Multiderivative Time Integrators for Hyperbolic Conservation Laws2013 •
SIAM Journal on Numerical Analysis
Internal Error Propagation in Explicit Runge--Kutta Methods2014 •
2012 •
Journal of Modern Methods in Numerical Mathematics
On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs2017 •
2014 •
Acta Universitatis Apulensis
On Runge–Kutta–Nystrom Formulae with Contractivity Preserving Properties for Second Order OdesInternational Journal for Numerical Methods in Fluids
High‐Order Implicit Time‐stepping with High‐order CENO Methods for Unsteady Three‐Dimensional CFD SimulationsJournal of Scientific computing
Implicitexplicit RungeKutta schemes and applications to hyperbolic systems with relaxation2005 •
21st AIAA Computational Fluid Dynamics Conference
Low-storage IMEX Runge-Kutta schemes for the simulation of Navier-Stokes systems2013 •
Journal of Computational Physics
Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge–Kutta time discretizations2007 •
2006 •
2003 •
2012 •
Journal of Computational Physics
Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection–diffusion equations on triangular meshes2013 •
Arxiv preprint arXiv:1010.1472
Exponential Runge-Kutta methods for stiff kinetic equations2010 •
Journal of Computational Physics
Time step restrictions for Runge–Kutta discontinuous Galerkin methods on triangular grids2008 •
2006 •
International Journal of Computer Mathematics - IJCM
STRONG STABILITY PRESERVING PROPERTY OF THE DEFERRED CORRECTION TIME DISCRETIZATION2008 •
Applied Mathematics and Computation
High order accurate semi-implicit WENO schemes for hyperbolic balance laws2011 •