Semi-analytic shooting methods for Burgers’ equation

Abstract

We implement new semi-analytic shooting methods for the stationary viscous Burgers’ equation by modifying the classical time differencing methods. When the viscosity is small, a very stiff boundary layer appears and this boundary layer causes significant difficulties to approximate the solution for Burgers’ equation. To overcome this issue and improve the numerical quality of the shooting methods with the classical Integrating Factor (IF) methods and Exponential Time Differencing (ETD) methods, we first employ the singular perturbation analysis for Burgers’ equation, and derived the so-called correctors that approximate the stiff part of the solution. Then, we build our new semi-analytic shooting methods for the stationary viscous Burgers’ equation by embedding these correctors into the IF and ETD methods. By performing numerical simulations, we verify that our new schemes, enriched with the correctors, give much better approximations, compared with the classical schemes.