Solving Burgers' Equation Using Optimal Rational Approximations

Abstra t. We solve vis ous Burger's equation using a fast and a urate algorithm referred to here as the redu tion algorithm for omputing near optimal rational approximations. Given a proper rational fun tion with n poles, the redu tion algorithm omputes (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 mu h more e ient than linear representations of fun tions via a xed basis (e.g. wavelets), their use in numeri al omputations has been limited by a la k of e ient, robust, and a urate algorithms. The redu tion algorithm presented here omputes reliably (near) optimal rational approximations with high a ura y (e.g., ≈ 10) and a omplexity that is essentially linear in the number n of original poles. A key tool is a re ently developed algorithm for omputing small on-eigenvalues of Cau hy matri es with high relative a ura y, an impossible task for standard algorithms without extended pre ision. Using the redu tion algorithm, we develop a numeri al al ulus for rational representations of fun tions. Indeed, while operations su h as multipli ation and onvolution in rease the number of poles in the representation, we use the redu tion algorithm to maintain an optimally small number of poles. To demonstrate the e ien y, robustness, and a ura y of our approa h, we solve Burgers' equation with small vis osity 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 apture the solutions with high a ura y using a small number of poles. In parti ular, we solve the equation with lo al a ura y ǫ = 10 for vis osity as small as ν = 10.