A Splitting Approach for Freezing Waves

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 longtime forward simulation. A difficulty with longtime 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 and Thümmler J Appl Dyn Syst 3:85–116, 2004, [2]), see also (Beyn et al. Current challenges in stability issues for numerical differential equations, 2014, [1]), 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 which is 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 longtime 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.