We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such regime the system relaxes towards a convection-diffusion equation. The first objective of the paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of the paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by an original reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis on such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, under the same conditions derived for hyperbolic relaxation [8]. Several numerical examples including neutron transport equations confirm the theoretical analysis.

In this paper we present an error analysis of the IMEX Runge-Kutta methods when applied to stiff problems containing a nonstiff term and a stiff term, characterized by a small stiffness parameter ε. In this analysis we expand the global error in powers of ε and show that the coefficients of the error are the global errors of the IMEX Runge-Kutta method applied to a differential-algebraic system. Interesting convergence results of these errors and of the remainder of the expansion allow us to determine sharp error bounds for stiff problems. As a representative example of stiff problems we have chosen the van der Pol equation. We illustrate that the theoretical prediction is confirmed by the numerical test. Specifically, an order reduction phenomenon is observed when the problem becomes increasingly stiff. In particular, making several assumptions, we try to improve global error estimates of several IMEX Runge-Kutta methods existing in the literature.

The main purpose of the paper is to show how to use implicit-explicit (IMEX) Runge-Kutta methods in a much more general context than usually found in the literature, obtaining very effective schemes for a large class of problems. This approach gives a great flexibility, and allows, in many cases the construction of simple linearly implicit schemes without any Newton's iteration. This is obtained by identifying the (possibly linear) dependence on the unknown of the system which generates the stiffness. Only the stiff dependence is treated implicitly, then making the whole method much simpler than fully implicit ones. The resulting schemes are denoted as semi-implicit R-K. We adopt several semi-implicit R-K methods up to order three. We illustrate the effectiveness of the new approach with many applications to reaction-diffusion, convection diffusion and nonlinear diffusion system of equations.

In this paper we consider hyperbolic systems with relaxation in which the relaxation time ε may vary from values of order one to very small values. When ε is very small, the relaxation term becomes very strong and highly stiff, and underresolved numerical schemes may produce spurious results. In such cases it is important to have schemes that work uniformly with respect to ε. IMEX R-K schemes have been widely used for the time evolution of hyperbolic partial differential equations but the schemes existing in literature do not exhibit uniform accuracy with respect to the relaxation time. We develop new Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with relaxation that present better uniform accuracy than the ones existing in the literature and in particular produce good behavior with high order accuracy in the asymptotic limit, i.e, when ε is very small. These schemes are obtained by imposing new additional order conditions to guarantee better accuracy over a wide range of the relaxation time. We propose the construction of new third-order IMEX R-K schemes of type CK [2]. In several test problems, these schemes, with a fixed spatial discretization, exhibit for all range of the relaxation time an almost uniform third-order accuracy.

In this paper we consider the development of Implicit-Explicit (IMEX) Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such systems the scaling depends on an additional parameter which modifies the nature of the asymptotic behavior which can be either hyperbolic or parabolic. Because of the multiple scalings, standard IMEX Runge-Kutta methods for hyperbolic systems with relaxation loose their efficiency and a different approach should be adopted to guarantee asymptotic preservation in stiff regimes. We show that the proposed approach is capable to capture the correct asymptotic limit of the system independently of the scaling used. Several numerical examples confirm our theoretical analysis.