DOI: 10.1007/s10915-004-4635-5 Journal of Scientific Computing, Vol. 25, Nos. 1/2, November 2005 (© 2005) On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations Sigal Gottlieb 1 Received October 1, 2003; accepted (in revised form) March 15, 2004 Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic par- tial differential equations. These high order time discretization methods pre- serve the strong stability properties–in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which con- nects the timestep restriction on SSP methods with the theory of monotonic- ity and contractivity. Optimal explicit SSP Runge–Kutta methods for nonlinear problems and for linear problems as well as implicit Runge–Kutta methods and multi step methods will be collected. KEY WORDS: Strong stability preserving; Runge–Kutta methods; multi step methods; high order accuracy; time discretization. AMS (MOS) SUBJECT CLASSIFICATION: 65M20; 65L06. 1. INTRODUCTION TO SSP METHODS In the use of numerical methods for approximating solutions of PDEs we typically rely on linear stability theory to guarantee convergence. The celebrated Lax equivalence theorem (see [34] Theorem 1.5.1) states that for a linear method consistent with a linear problem, stability is neces- sary and sufficient for convergence. Strang [33] extended this result, and showed that for nonlinear problems if an approximation is consistent and its linearized version is L 2 stable, then for sufficiently smooth problems this approximation is convergent. However, solutions of hyperbolic partial differential equations (PDEs) are frequently discontinuous. In this case, 1 Department of Mathematics, University of Massachusetts Dartmouth, USA. E-mail: sg@cfm.brown.edu 105 0885-7474/05/1100-0105/0 © 2005 Springer Science+Business Media, Inc.