A splitting approach for freezing waves

TL;DR

This paper introduces a novel operator splitting numerical method for accurately approximating traveling waves in hyperbolic-parabolic PDE systems by transforming the problem into a PDAE and solving it with tailored splitting schemes.

Contribution

The authors develop a new splitting-based discretization method for the freezing PDAE, enabling efficient long-time simulation of traveling waves in coupled hyperbolic-parabolic systems.

Findings

Achieved linear and quadratic convergence rates with Lie- and Strang-splitting.

Successfully tested the method on viscous Burgers' equation.

Demonstrated the scheme's potential for hyperbolic-parabolic wave problems.

Abstract

We present a numerical method which is able to approximate traveling waves (e.g. viscous profiles) in systems with hyperbolic and parabolic parts by a direct long-time forward simulation. A difficulty with long-time simulations of traveling waves is that the solution leaves any bounded computational domain in finite time. To handle this problem one should go into a suitable co-moving frame. Since the velocity of the wave is typically unknown, we use the method of freezing [Beyn, Th\"ummler 2004], see also [Beyn, Otten, Rottmann-Matthes 2014], which transforms the original partial differential equation (PDE) into a partial differential algebraic equation (PDAE) and calculates a suitable co-moving frame on the fly. The efficient numerical approximation of this freezing PDAE is a challenging problem and we introduce a new numerical discretization, suitable for problems that consist of hyperbolic conservation laws which are (nonlinearly) coupled to parabolic equations. The idea of our scheme is to use the operator splitting approach. The benefit of splitting methods in this context lies in the possibility to solve hyperbolic and parabolic parts with different numerical algorithms. We test our method at the (viscous) Burgers' equation. Numerical experiments show linear and quadratic convergence rates for the approximation of the numerical steady state obtained by a long-time simulation for Lie- and Strang-splitting, respectively. Due to these affirmative results we expect our scheme to be suitable also for freezing waves in hyperbolic-parabolic coupled PDEs.