The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Total-variation-diminishing time discretizations
SIAM Journal on Scientific and Statistical Computing
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Runge-Kutta research in Trondheim
Applied Numerical Mathematics - Special issue celebrating the centenary of Runge-Kutta methods
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
Positivity of Runge-Kutta and diagonally split Runge-Kutta methods
Applied Numerical Mathematics - Selected papers on eighth conference on the numerical treatment of differential equations 1-5 September 1997, Alexisbad, Germany
Linearly implicit Runge—Kutta methods for advection—reaction—diffusion equations
Applied Numerical Mathematics
A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods
SIAM Journal on Numerical Analysis
Non-linear evolution using optimal fourth-order strong-stability-preserving Runge-Kutta methods
Mathematics and Computers in Simulation - Nonlinear waves: computation and theory II
On Strong Stability Preserving Time Discretization Methods
Journal of Scientific Computing
Representations of Runge-Kutta Methods and Strong Stability Preserving Methods
SIAM Journal on Numerical Analysis
On High Order Strong Stability Preserving Runge---Kutta and Multi Step Time Discretizations
Journal of Scientific Computing
Implicit---Explicit Runge---Kutta Schemes and Applications to Hyperbolic Systems with Relaxation
Journal of Scientific Computing
Stepsize Conditions for General Monotonicity in Numerical Initial Value Problems
SIAM Journal on Numerical Analysis
A Numerical Study of Diagonally Split Runge---Kutta Methods for PDEs with Discontinuities
Journal of Scientific Computing
Optimal implicit strong stability preserving Runge--Kutta methods
Applied Numerical Mathematics
High Order Strong Stability Preserving Time Discretizations
Journal of Scientific Computing
Characterizing Strong Stability Preserving Additive Runge-Kutta Methods
Journal of Scientific Computing
Step Sizes for Strong Stability Preservation with Downwind-Biased Operators
SIAM Journal on Numerical Analysis
Binary weighted essentially non-oscillatory (BWENO) approximation
Journal of Computational and Applied Mathematics
Journal of Computational Physics
| Hi-index | 0.01 |
This paper deals with the numerical solution of initial value problems, for systems of ordinary differential equations, by Runge-Kutta methods (RKMs) with special nonlinear stability properties indicated by the terms total-variation-diminishing (TVD), strongly stable and monotonic. Stepsize conditions, guaranteeing these properties, were studied earlier, see e.g. Shu and Osher [C.-W. Shu, S. Osher, J. Comput. Phys. 77 (1988) 439-471], Gottlieb et al. [S. Gottlieb, C.-W. Shu, E. Tadmor, SIAM Rev. 43 (2001) 89-112], Hundsdorfer and Ruuth [W.H. Hundsdorfer, S.J. Ruuth, Monotonicity for time discretizations, in: D.F. Griffiths, G.A. Watson (Eds.), Proc. Dundee Conference 2003, Report NA/217, Univ. Dundee, 2003, pp. 85-94], Higueras [I. Higueras, J. Sci. Computing 21 (2004) 193-223; I. Higueras, SIAM J. Numer. Anal. 43 (2005) 924-948], Gottlieb [S. Gottlieb, J. Sci. Computing 25 (2005) 105-128], Ferracina and Spijker [L. Ferracina, M.N. Spijker, SIAM J. Numer. Anal. 42 (2004) 1073-1093; L. Ferracina, M.N. Spijker, Math. Comp. 74 (2005) 201-219]. Special attention was paid to RKMs which are optimal, in that the corresponding stepsize conditions are as little restrictive as possible within a given class of methods. Extensive searches for such optimal methods were made in classes of explicit RKMs, see e.g. Gottlieb and Shu [S. Gottlieb, C.-W. Shu, Math. Comp. 67 (1998) 73-85], Spiteri and Ruuth [R.J. Spiteri, S.J. Ruuth, SIAM J. Numer. Anal. 40 (2002) 469-491; R.J. Spiteri, S.J. Ruuth, Math. Comput. Simulation 62 (2003) 125-135], Ruuth [S.J. Ruuth, Math. Comp. 75 (2006) 183-207]. In the present paper we search for methods that are optimal in the above sense, within the interesting class of singly-diagonally-implicit Runge-Kutta (SDIRK) methods, with s stages and order p. Some methods, with 1==3) which we conjecture to be optimal for p=2 and p=3, respectively. Furthermore we prove, for strongly stable SDIRK methods, the order barrier p=