Steven J. Ruuth

Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations

Abstract Implicit-explicit (IMEX) linear multistep time-discretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable time-step restrictions when applied to convection-diffusion problems, unless diffusion strongly dominates and an appropriate BDF-based scheme is selected (Ascher et al., 1995). In this paper, we develop Runge-Kutta-based IMEX schemes that have better stability regions than the best known IMEX multistep schemes over a wide parameter range.

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

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.

Implicit-explicit methods for reaction-diffusion problems in pattern formation

Reaction-diffusion mechanisms have been used to explain pattern formation in developmental biology and in experimental chemical systems. To approximate the corresponding spatially discretized models, an explicit scheme can be used for the reaction term and an implicit scheme for the diffusion term. Such implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods in fluid flow problems.In this work, we analyze the performance of several of the best known linear multistep IMEX schemes for reaction-diffusion problems in pattern formation. For the linearized two chemical system, the growth properties exhibited by IMEX schemes are examined. Schemes which accurately represent the growth of the linearized problem for large time steps are identified. Numerical experiments show that first order accurate schemes, as well as schemes which produce only a weak decay of high frequency spatial error may yield plausible results which are nonetheless qualitatively incorrect. For such schemes, computations using refinements in the time step are likely to produce essentially the same (erroneous) results. Higher order schemes which produce a strong decay of high frequency errors are proposed instead.Our findings are demonstrated on several examples.

Implicit-Explicit Methods for Time-Dependent PDE''s

Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusion-convection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reaction-diffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods. For the prototype linear advection-diffusion equation, a stability analysis for first, second, third and fourth order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified. Numerical experiments demonstrate that weak decay of high frequency modes can lead to extra iterations on the finest grid when using multigrid computations with finite difference spatial discretization, and to aliasing when using spectral collocation for spatial discretization. When this behaviour occurs, use of weakly damping schemes such as the popular combination of Crank-Nicolson with second order Adams-Bashforth is discouraged and better alternatives are proposed. Our findings are demonstrated on several examples.

Global optimization of explicit strong-stability-preserving Runge-Kutta methods

Strong-stability-preserving Runge-Kutta (SSPRK) methods are a type of time discretization method that are widely used, especially for the time evolution of hyperbolic partial differential equations (PDEs). Under a suitable stepsize restriction, these methods share a desirable nonlinear stability property with the underlying PDE; e.g., positivity or stability with respect to total variation. This is of particular interest when the solution exhibits shock-like or other nonsmooth behaviour. A variety of optimality results have been proven for simple SSPRK methods. However, the scope of these results has been limited to low-order methods due to the detailed nature of the proofs. In this article, global optimization software, BARON, is applied to an appropriate mathematical formulation to obtain optimality results for general explicit SSPRK methods up to fifth-order and explicit low-storage SSPRK methods up to fourth-order. Throughout, our studies allow for the possibility of negative coefficients which correspond to downwind-biased spatial discretizations. Guarantees of optimality are obtained for a variety of third- and fourth-order schemes. Where optimality is impractical to guarantee (specifically, for fifth-order methods and certain low-storage methods), extensive numerical optimizations are carried out to derive numerically optimal schemes. As a part of these studies, several new schemes arise which have theoretically improved time-stepping restrictions over schemes appearing in the recent literature.

Monotonicity-Preserving Linear Multistep Methods

In this paper we provide an analysis of monotonicity properties for linear multistep methods. These monotonicity properties include positivity and the diminishing of total variation. We also pay particular attention to related boundedness properties such as the total variation bounded (TVB) property. In the analysis the multistep methods are considered in combination with suit- able starting procedures. This allows for monotonicity statements for classes of methods which are important and often used in practice but which were thus far not covered by theoretical results.

Two Barriers on Strong-Stability-Preserving Time Discretization Methods

Strong-stability-preserving (SSP) time discretization methods are popular and effective algorithms for the simulation of hyperbolic conservation laws having discontinuous or shock-like solutions. They are (nonlinearly) stable with respect to general convex functionals including norms such as the total-variation norm and hence are often referred to as total-variation-diminishing (TVD) methods. For SSP Runge–Kutta (SSPRK) methods with positive coefficients, we present results that fundamentally restrict the achievable CFL coefficient for linear, constant-coefficient problems and the overall order of accuracy for general nonlinear problems. Specifically we show that the maximum CFL coefficient of an s-stage, order-p SSPRK method with positive coefficients is s−p+1 for linear, constant-coefficient problems. We also show that it is not possible to have an s-stage SSPRK method with positive coefficients and order p>4 for general nonlinear problems.

High-Order Strong-Stability-Preserving Runge-Kutta Methods with Downwind-Biased Spatial Discretizations

Strong-stability-preserving Runge--Kutta (SSPRK) methods are a specific type of time discretization method that have been widely used for the time evolution of hyperbolic partial differential equations (PDEs). Under a suitable stepsize restriction, these methods share a desirable nonlinear stability property with the underlying PDE, e.g., stability with respect to total variation, the maximum norm, or other convex functionals. This is of particular interest when the solution exhibits shock-like or other nonsmooth behavior. Many results are known for SSPRK methods with nonnegative coefficients. However, it has recently been shown that such methods cannot exist with order greater than 4. In this paper, we give a systematic treatment of explicit SSPRK methods with general (i.e., possibly negative) coefficients up to order 5. In particular, we show how to optimally treat negative coefficients (corresponding to a change in the upwind direction of the spatial discretization) in the context of effective CFL coefficient maximization and provide proofs of optimality of some explicit SSPRK methods of orders 1 to 4. We also give the first known explicit fifth-order SSPRK schemes and show their effectiveness in practice versus more well-known fifth-order explicit Runge--Kutta schemes.

Optimal Strong-Stability-Preserving Time-Stepping Schemes with Fast Downwind Spatial Discretizations

In the field of strong-stability-preserving time discretizations, a number of researchers have considered using both upwind and downwind approximations for the same derivative, in order to guarantee that some strong stability condition will be preserved. The cost of computing both the upwind and downwind operator has always been assumed to be double that of computing only one of the two. However, in this paper we show that for the weighted essentially non-oscillatory method it is often possible to compute both these operators at a cost that is far below twice the cost of computing only one. This gives rise to the need for optimal strong-stability-preserving time-stepping schemes which take into account the different possible cost increments. We construct explicit linear multistep schemes up to order six and explicit Runge–Kutta schemes up to order four which are optimal over a range of incremental costs

A Numerical Study of Diagonally Split Runge---Kutta Methods for PDEs with Discontinuities

Abstract 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.