An exponential time-differencing method for monotonic relaxation systems

Abstract

Abstract. We present first and second-order accurate exponential time differencing methods for a special class of stiff ODEs, denoted as monotonic relaxation ODEs. Some desirable accuracy and robustness properties of our methods are established. In particular, we prove a strong form of stability denoted as monotonic asymptotic stability, guaranteeing that no overshoots of the equilibrium value are possible. This is motivated by the desire to avoid spurious unphysical values that could crash a large simulation.We present a simple numerical example, demonstrating the potential for increased accuracy and robustness compared to established Runge-Kutta and exponential methods. Through operator splitting, an application to granulargas flow is provided.

Arbitrary-rate relaxation techniques for the numerical modeling of compressible two-phase flows with heat and mass transfer

“…Our idea consists in describing the relaxation processes by systems of ordinary equations obtained from the governing two-phase equations that admit analytical semi-exact exponential solutions. Similar approaches using exponential solutions to solve stiff relaxation systems were used for instance in [20,26,52,3,16,17]. Let us remark some differences with respect to our previous work [17,18] on relaxation techniques for noninstantaneous heat and mass transfers and general equation of state.…”

Shock–contact–shock solutions of the Riemann problem for dilute granular gas

“…A slightly different problem, though its solution has similar features to the piston problem, is the wall collision of granular gas falling under the action of gravity (Pareschi & Russo 2005; Serna & Marquina 2005; Kamath & Du 2009; Aursand et al. 2014).…”

Splitting methods for relaxation two-phase flow models

“…The relaxation ODE under consideration in this work is of this type, and it is easy to verify that for such ODEs the backward Euler scheme will be stable in the sense that the solution to (22) will not overshoot an equilibrium point (Aursand et al, 2010). However, it should be emphasized that obtaining this solution requires solving a non-linear system of equations (22) by an iterative scheme such as the Newton-Raphson method.…”

“…The basic idea is that one gets rid of stability restrictions on the time step by approximating the stiff component of the solution as an exponential function. Recently, exponential methods tailored for relaxation systems have been proposed (Aursand et al, 2010). The first order method, referred to as ASY1, is given by…”