Solving Burgersʼ equation using optimal rational approximations

Abstract

We solve viscous Burgersʼ equation using a fast and accurate algorithm—referred to here as the reduction algorithm—for computing near optimal rational approximations.Given a proper rational function with n poles, the reduction algorithm computes (for a desired L∞-approximation error) a rational approximation of the same form, but with a (near) optimally small number m≪n of poles. Although it is well known that (nonlinear) optimal rational approximations are much more efficient than linear representations of functions via a fixed basis (e.g. wavelets), their use in numerical computations has been limited by a lack of efficient, robust, and accurate algorithms. The reduction algorithm presented here computes reliably (near) optimal rational approximations with high accuracy (e.g., ≈10−14) and a complexity that is essentially linear in the number n of original poles. A key tool is a recently developed algorithm for computing small con-eigenvalues of Cauchy matrices with high relative accuracy, an impossible task for standard algorithms without extended precision.Using the reduction algorithm, we develop a numerical calculus for rational representations of functions. Indeed, while operations such as multiplication and convolution increase the number of poles in the representation, we use the reduction algorithm to maintain an optimally small number of poles.To demonstrate the efficiency, robustness, and accuracy of our approach, we solve Burgersʼ equation with small viscosity ν. It is well known that its solutions exhibit moving transition regions of width O(ν), so that this equation provides a stringent test for adaptive PDE solvers. We show that optimal rational approximations capture the solutions with high accuracy using a small number of poles. In particular, we solve the equation with local accuracy ϵ=10−9 for viscosity as small as ν=10−5.