Vu Thai Luan

Vu Thai Luan, Dominik L. Michels

We introduce a class of explicit exponential Rosenbrock methods for the time integration of large systems of stiff differential equations. Their application with respect to simulation tasks in the field of visual computing is discussed where these time integrators have shown to be very competitive compared to standard techniques. In particular, we address the simulation of elastic and nonelastic deformations as well as collision scenarios focusing on relevant aspects like stability and energy conservation, large stiffnesses, high fidelity and visual accuracy.

Vu Thai Luan, Mayya Tokman, Greg Rainwater

We propose two new classes of time integrators for stiff DEs: the implicit-explicit exponential (IMEXP) and the hybrid exponential methods. In contrast to the existing exponential schemes, the new methods offer significant computational advantages when used with preconditioners. Any preconditioner can be used with any of these new schemes. This leads to a broader applicability of exponential methods. The proof of stability and convergence of these integrators and numerical demonstration of their efficiency are presented.

Vu Thai Luan

Among the family of fourth-order time integration schemes, the two-stage Gauss--Legendre method, which is an implicit Runge--Kutta method based on collocation, is the only superconvergent. The computational cost of this implicit scheme for large systems, however, is very high since it requires solving a nonlinear system at every step. Surprisingly, in this work we show that one can construct and prove convergence results for exponential methods of order four which use two stages only. Specifically, we derive two new fourth-order two-stage exponential Rosenbrock schemes for solving large systems of differential equations. Moreover, since the newly schemes are not only superconvergent but also fully explicit, they clearly offer great advantages over the two-stage Gauss--Legendre method as well as other time integration schemes. Numerical experiments are given to demonstrate the efficiency of the new integrators.

Vu Thai Luan, Rujeko Chinomona, Daniel R. Reynolds

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step exponential integrators. More precisely, starting from an explicit exponential Runge--Kutta method of the appropriate form, we derive a multirate algorithm to approximate the action of the matrix exponential through the definition of modified "fast" initial-value problems. These fast problems may be solved using any viable solver, enabling multirate simulations through use of a subcycled method. Due to this structure, we name these Multirate Exponential Runge--Kutta (MERK) methods. In addition to showing how MERK methods may be derived, we provide rigorous convergence analysis, showing that for an overall method of order $p$, the fast problems corresponding to internal stages may be solved using a method of order $p-1$, while the final fast problem corresponding to the time-evolved solution must use a method of order $p$. Numerical simulations are then provided to demonstrate the convergence and efficiency of MERK methods with orders three through five on a series of multirate test problems.

Vu Thai Luan, Janusz A. Pudykiewicz, Daniel R. Reynolds

In this paper, we investigate the use of higher-order exponential Rosenbrock time integration methods on the shallow water equations on the sphere. This stiff, nonlinear model provides a testing ground for accurate and stable time integration methods in weather modeling, serving as the focus for exploration of novel methods for many years. We therefore identify a candidate set of three recent exponential Rosenbrock methods of orders four and five (exprb42, pexprb43 and exprb53) for use on this model. Based on their multi-stage structure, we propose a set of modifications to the phipm_IOM2 algorithm for efficiently calculating the matrix phi-functions. We then investigate the performance of these methods on a suite of four challenging test problems, comparing them against the epi3 method investigated previously in [1, 2] on these problems. In all cases, the proposed methods enable accurate solutions at much longer time-steps than epi3, proving considerably more efficient as either the desired solution error decreases, or as the test problem nonlinearity increases.