On stability issues for IMEX schemes applied to 1D scalar hyperbolic equations with stiff reaction terms

Abstract

Abstract. The application of a Method of Lines to a hyperbolic PDE with source terms gives rise to a system of ODEs containing terms that may have very different stiffness properties. In this case, Implicit-Explicit Runge-Kutta (IMEX-RK) schemes are particularly useful as high order time integrators because they allow an explicit handling of the convective terms, which can be discretized using the highly developed shock capturing technology, together with an implicit treatment of the source terms, necessary for stability reasons.Motivated by the structure of the source term in a model problem introduced by LeVeque and Yee in [J. Comput. Phys. 86 (1990)], in this paper we study the preservation of certain invariant regions as a weak stability constraint. For the class of source terms considered in this paper, the unit interval is an invariant region for the model balance law. In the first part of the paper, we consider first order time discretizations, which are the basic building blocks of higher order IMEX-RK schemes, and study the conditions that guarantee that [0, 1] is also an invariant region for the numerical scheme. In the second part of the paper, we study the conditions that ensure the preservation of this property for higher order IMEX schemes.

A unsplitting finite volume method for models with stiff relaxation source terms

“…Motivated by the stability of IMEX-methods [10,22], we use the robust trapezoidal rule in a convenient way,…”

“…Thus, a very convenient strategy to analyse this class of problems is to make use of accurate numerical methods in order to account the underlying set of solutions in equilibrium and non-equilibrium regimes. In the recent years, prominent wellbalanced and asymptotic preserving schemes were developed for solving balance laws, in particular when G(U ) = U , e.g., [3,4,6,10,11,12,13,16,21,22]). Essentially, a method is said to be well-balanced if it preserves an unperturbed steady state in a such way that properly balance the fluxes and the source at the discrete level.…”

Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws

“…For example, when the source term is a non-increasing function, the total variation of the exact solution of the scalar balance law is also a nonincreasing function, as in the homogeneous case (see, e.g., [16,10]). In general, however, the source term might not be decreasing (see [6,11,27]) and some semiimplicit and fully implicit scheme are not applicable, at least in a straightforward manner [6,11].…”

A posteriori error analysis of IMEX multi-step time integration methods for advection–diffusion–reaction equations

“…A set of representative references from the scientific literature indicates both the recent interest in these methods, as well as the complexity of the computational physics/mathematical models to which these methods have been applied (see e.g. [2][3][4][5][6][7][8][9][10][11][12]). Specific references that consider multi-step IMEX approaches for complex applications include [13][14][15][16].…”

“…4 Errors and effectivity ratios for the example in Section 5.2.4. Errors for different choices of f and g for the example in Section 5.2.5 for first-order SBDF scheme.…”