An IMEX-RK scheme for capturing similarity solutions in the multidimensional Burgers’ equation

In this paper we introduce a new, simple and efficient numerical scheme for the implementation of the freezing method for capturing similarity solutions in partial differential equations. The scheme is based on an IMEX-Runge-Kutta approach for a method of lines (semi-)discretization of the freezing partial differential algebraic equation (PDAE). We prove second order convergence for the time discretization at smooth solutions in the ODE-sense and we present numerical experiments that show second order convergence for the full discretization of the PDAE. As an example serves the multi-dimensional Burgers’ equation. By considering very different sizes of viscosity, Burgers’ equation can be considered as a prototypical example of general coupled hyperbolicparabolic PDEs. Numerical experiments show that our method works perfectly well for all sizes of viscosity, suggesting that the scheme is indeed suitable for capturing similarity solutions in general hyperbolic-parabolic PDEs by direct forward simulation with the freezing method

An IMEX-RK scheme for capturing similarity solutions in the multidimensional Burgers’ equation

In this paper we introduce a new, simple and efficient numerical scheme for the implementation of the freezing method for capturing similarity solutions in partial differential equations. The scheme is based on an IMEX-Runge-Kutta approach for a method of lines (semi-)discretization of the freezing partial differential algebraic equation (PDAE). We prove second order convergence for the time discretization at smooth solutions in the ODE-sense and we present numerical experiments that show second order convergence for the full discretization of the PDAE. As an example serves the multi-dimensional Burgers’ equation. By considering very different sizes of viscosity, Burgers’ equation can be considered as a prototypical example of general coupled hyperbolicparabolic PDEs. Numerical experiments show that our method works perfectly well for all sizes of viscosity, suggesting that the scheme is indeed suitable for capturing similarity solutions in general hyperbolic-parabolic PDEs by direct forward simulation with the freezing method

Freezing similarity solutions in multi-dimensional Burgers’ Equation

The topic of this paper are similarity solutions occurring in multi-dimensional Burgers’ equation. We present a simple derivation of the symmetries appearing in a family of generalizations of Burgers’ equation in d-space dimensions. These symmetries we use to derive an equivalent partial differential algebraic equation (freezing system) that allows us to do long time simulations and obtain good approximations of similarity solutions by direct forward simulation. The method also allows us without further effort to observe meta-stable behavior near N-wave-like patterns

Freezing similarity solutions in multi-dimensional Burgers’ Equation

The topic of this paper are similarity solutions occurring in multi-dimensional Burgers’ equation. We present a simple derivation of the symmetries appearing in a family of generalizations of Burgers’ equation in d-space dimensions. These symmetries we use to derive an equivalent partial differential algebraic equation (freezing system) that allows us to do long time simulations and obtain good approximations of similarity solutions by direct forward simulation. The method also allows us without further effort to observe meta-stable behavior near N-wave-like patterns

A splitting approach for freezing waves

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 [2], see also [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, 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

Freezing traveling and rotatingwaves in second order evolution equations

In this paper we investigate the implementation of the so-called freezing method for second order wave equations in one and several space dimensions. The method converts the given PDE into a partial differential algebraic equation which is then solved numerically. The reformulation aims at separating the motion of a solution into a co-moving frame and a profile which varies as little as possible. Numerical examples demonstrate the feasability of this approach for semilinear wave equations with sufficient damping. We treat the case of a traveling wave in one space dimension and of a rotating wave in two space dimensions. In addition, we investigate in arbitrary space dimensions the point spectrum and the essential spectrum of operators obtained by linearizing about the profile, and we indicate the consequences for the nonlinear stability of the wave

Computation and Stability of TravelingWaves in Second Order Evolution Equations

The topic of this paper are nonlinear traveling waves occuring in a system of damped waves equations in one space variable. We extend the freezing method from first to second order equations in time. When applied to a Cauchy problem, this method generates a comoving frame in which the solution becomes stationary. In addition it generates an algebraic variable which converges to the speed of the wave, provided the original wave satisfies certain spectral conditions and initial perturbations are sufficiently small. We develop a rigorous theory for this effect by recourse to some recent nonlinear stability results for waves in first order hyperbolic systems. Numerical computations illustrate the theory for examples of Nagumo and FitzHugh-Nagumo type

Computation and stability of patterns in hyperbolic-parabolic Systems

Rottmann-Matthes J. Computation and stability of patterns in hyperbolic-parabolic Systems. Berichte aus der Mathematik. Aachen: Shaker; 2010

Resolvent estimates for boundary value problems on large intervals via the theory of discrete approximations

Beyn W-J, Rottmann-Matthes J. Resolvent estimates for boundary value problems on large intervals via the theory of discrete approximations. Numerical Functional Analysis and Optimization . 2007;28(5-6):603-629.In many applications such as the stability analysis of traveling waves, it is important to know the spectral properties of a linear differential operator on the whole real line. We investigate the approximation of this operator and its spectrum by finite interval boundary value problems from an abstract point of view. Under suitable assumptions on the boundary operators, we prove that the approximations converge regularly (in the sense of discrete approximations) to the all line problem, which has strong implications for the behavior of resolvents and spectra. As an application, we obtain resolvent estimates for abstract coupled hyperbolic - parabolic equations. Furthermore, we show that our results apply to the FitzHugh - Nagumo system