Hybrid Numerical Schemes For Convection-diffusion Equation

Abstract: In this talk, we present higher order hybrid BSQI-WENO schemes for the solution of convection-diffusion problem. In the BSQI-WENO schemes, B-spline quasi-interpolation based numerical schemes are conjugated with the fifth order weighted essentially non-oscillatory (WENO) scheme. A weak local residual based estimate is used to detect the discontinuities and high gradient regions of the solution. In smooth regions, we use the BSQI approximation to the derivative of the unknown and in the shock (high gradient) regions, we use the WENO scheme. For the time discretization, strong stability preserving Runge-Kutta of order three is considered. The accuracy of the proposed scheme is demonstrated by numerical experiments. We also discuss the hybrid scheme of arbitrary order for hyperbolic conservation laws and hybrid finite difference methods for porous medium equation.