IMEX extensions of linear multistep methods with general monotonicity and boundedness properties

Abstract

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationIMEX extensions of linear multistep methods with general monotonicity and boundedness properties ABSTRACT For solving hyperbolic systems with stiff sources or relaxation terms, time stepping methods should combine favorable monotonicity properties for shocks and steep solution gradients with good stability properties for stiff terms. In this paper we consider implicit-explicit (IMEX) multistep methods. Suitable methods will be constructed, based on explicit methods with general monotonicity and boundedness properties for hyperbolic equations. Numerical comparisons are made with several implicit-explicit Runge-Kutta methods.

$$L^\infty $$ -Stability of IMEX-BDF2 Finite Volume Scheme for Convection-Diffusion Equation

“…This section is devoted to the design of a numerical scheme to approximate (2), using the vertex-based MUSCL finite volume methods introduced in [13] and used in [2] for a second-order accuracy in space, and an implicit-explicit (IMEX) linear multistep methods [10] for a second-order in time.…”

“…For the time discretization, we consider the implicit BDF2 scheme and an extrapolated BDF2 scheme following [10]. With this choice, we obtain a second-order accuracy in time.…”

A relaxation method for calculating detonation in condensed explosives

“…A fifth-order IMEX scheme of linear multistep methods with general monotonicity and boundedness properties [9] is adopted to discretize the ordinary differential equations (7): Finally, the relaxation scheme with temporal-spatial fifth-order precision about the detonation flows in condensed explosives turns into the expression (8). It is worthy of indicating that the discretization procedure does not solve Riemann problem.…”

“…After the nonlinear governing equations of the condensed explosives are transformed into linear relaxation equations, an improved fifth-order weighted essentially nonoscillatory (WENO) [19] is utilized to spatially discretize and a fifth-order IMEX scheme of linear multistep methods with general monotonicity and boundedness properties is utilized to temporally discretize [9]. The numerical example about one-dimensional steady structure of detonation wave in PBX-9404 demonstrates that our method has high accuracy and high resolution properties.…”

“…In order to exactly capturing the discontinuity, besides the high resolution in spatial discretization, the high resolution in temporal discretization is very necessary. Representative results [9] show that the unsplitting explicit and implicit scheme is much more reliable: the advection term adopts explicit scheme, and the source term adopts implicit scheme.…”