Numerical Simulation of Burgers Flow by Wavelet Method

The paper is concerned with a numerical simulation of Burgers flow as a simplified model of unsteady flow of a compressible viscous fluid. Viscous Burgers equation represents many of the properties of unsteady compressible NavierStokes equations, such as nonlinear convection and viscous diffusion leading to shock waves and boundary layers. A one-dimensional Burgers equation is frequently used to test new methods because an analytic solutions are known for different boundary and initial conditions. We follow some ideas from [1, 3, 5], where a general wavelet adaptive method for a large class of nonlinear equations has been proposed and we solve Burgers equation by a wavelet adaptive method. The adaptivity in the context of wavelet discretization insists in establishing which wavelet coefficients to keep and which to discard. The specific difficulty is that the singularities might move in time and so the set of indices of significant wavelet coefficients at each time step should be updated. The computational complexity for all steps of our algorithm is controlled. The computation is carried out in MATLAB using a PDE Toolbox and Wavelet Toolbox. 1 Motivation compression property of wavelets Let us approximate the function f by a combination fN = ∑ (j,k)∈JN cj,kψj,k, where #JN = N and ψj,k are suitable wavelets. A function f that is smooth, except at some isolated singularities, typically has a sparse representation in a wavelet basis, i.e. only a small number of numerically significant coefficients carry most of the information on f . Figure 1 displays a function f sampled on 29 points and its reconstruction from 50 largest wavelet coefficients. Wavelets used for decomposition are Daubechies wavelets with two vanishing moments. The largest coefficients are also displayed in Figure 1, the x-axis represents the center of the support of wavelet corresponding to given coefficient and y-axis represents the level of resolution (j). The function f has sharp derivative at the point x = 0.5 and so we can observe that the approximation is automatically refined near this point. This compression property of wavelets has many applications. Most important are data compression, signal analysis, and efficient adaptive schemes for PDE’s. Figure 1: function f 50 significant coefficients approximation fN of f 2 Viscous Burgers equation Let Ω = 〈a, b〉 be a bounded interval, T > 0 and QT := Ω× (0, T ). We consider viscous Burgers equation with homogeneous Dirichlet boundary conditions: Find u : QT → R such that ∂u ∂t + u ∂u ∂x = ν ∂2u ∂x2 in QT (1) with initial and boundary conditions u (x, 0) = u (x) for x ∈ Ω and u (a, t) = u (b, t) = 0 for t ∈ (0, T ). (2) We assume that ν > 0, u0 ∈ L2 (Ω). We also consider Burgers equation with periodic boundary conditions. Burgers equation is derived from the Navier-Stokes equation in the case of a one-dimensional non-stationary flow of a compressible viscous fluid and it models a flow through a shock wave. This equation is frequently used to test new methods because an analytic solutions are known for different boundary and initial conditions and because for small values of ν solutions typically develop very sharp gradients which are difficult to reproduce with numerical methods. Weak solution of viscous Burgers equation is defined as a function u satisfying: 1) u ∈ L2 (0, T ;H1 0 (Ω) ) , u ∈ L∞ (QT ) , 2) d dt 〈u (t) , v〉+ ∫