A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods

Abstract

Strong-stability-preserving (SSP) time discretization methods have a nonlinear stability property that makes them particularly suitable for the integration of hyperbolic conservation laws where discontinuous behavior is present. Optimal SSP schemes have been previously found for methods of order 1, 2, and 3, where the number of stages s equals the order p. An optimal low-storage SSP scheme with s = p = 3 is also known. In this paper, we present a new class of optimal high-order SSP and low-storage SSP Runge-Kutta schemes with s > p. We find that these schemes are ultimately more efficient than the known schemes with s = p because the increase in the allowable time step more than offsets the added computational expense per step. We demonstrate these efficiencies on a set of scalar conservation laws. AMS subject classifications. 65L06, 65M20PII. S0036142901389025 1. Introduction. The method of lines is a popular semidiscretization method for the solution of time-dependent partial differential equations (PDEs). The idea behind it is to first suitably discretize the spatial variables (e.g., by finite differences, finite volumes, finite elements, or spectral methods) to yield a set of ordinary differential equations (ODEs) in time. Then, this set of ODEs can be integrated using standard time-stepping techniques such as linear multistep or Runge-Kutta methods.Standard stability analysis for the solvers of such systems generally focuses on linear stability. Indeed, such analysis is often adequate when the desired solutions are smooth. However, solutions to hyperbolic PDEs may not be smooth: shock waves or other discontinuous behavior can develop even from smooth initial data. In such cases, standard discretizations based on linear stability analysis suffer from poor performance due to the presence of spurious oscillations, overshoots, and progressive smearing. The numerical solutions obtained from these discretizations often exhibit a weak form of instability (called nonlinear instability) resulting in unphysical behavior. Accordingly, numerical methods based on a nonlinear stability requirement are very desirable. Such methods were originally referred to as total variation diminishing (TVD) [17]; see also the subsequent articles [18,6]. However, following the more recent article [7], we refer to them in this paper as strong-stability-preserving (SSP) methods.We are interested in the development, implementation, and analysis of a new class of optimal SSP Runge-Kutta (SSPRK) time-stepping schemes for the system of

Computing multi-mode shock-induced compressible turbulent mixing at late times

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Two-Phase Flows of Droplets in Contractions and Double Bends

“…2, No. 3 (2008) derivatives has been chosen are solved using explicit total-variation diminishing (TVD) RungeKutta (RK) time integration (Osher and Fedkiw, 2003;Spiteri & Ruuth, 2002). In addition, the LSM implicitly provides the conditions for the revised, nearly second-order, two-phase, time-dependent, 2-D Navier-Stokes momentum equations' solver, incorporating viscous and surface tension terms depending on the level-set function.…”

Surprising computations

“…Much work was carried out in constructing such methods and in providing extensive nonlinear theory, see [13,33,20,18] and references therein. There is general agreement in the relevant community that the SSP concept has yielded a bullet-proof class of time discretization methods in conjunction with ENO, even though examples where the SSP property is actually essential for performance are uncommon and even though spurious oscillations using ENO remain possible in principle.…”