WENO Methods And The Applications In Convection Diffusion Equations

In this thesis, we study weighted essentially non-oscillatory (WENO) methods and their applications in convection-diffusion equations.WENO methods are relatively recent yet quite popular high order numerical meth-ods for solving convection dominated partial differential equations, in particular hy-perbolic conservation laws. The main advantage of such methods is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, non-oscillatory and sharp discontinuity transitions.The WENO idea is to choose the final approximation as a nonlinear convex com-bination of lower order approximations on candidate stencil, to obtain a higher order approximation. The WENO methods can be designed in the finite difference or finite volume framework. The WENO procedure can be used in different contexts, such as WENO interpolation, WENO reconstruction and WENO integration. The stability and non-oscillatory performance of the WENO procedure depend rather crucially on the positivity of the linear weights. We study on the positivity of linear weights in a few typical WENO procedures, and then present the explicit formulae and the positivity intervals for the linear weights, in order to lay a solid foundation for further design of various WENO schemes including WENO interpolation, WENO reconstruction and WENO approximation to first and second derivatives, and WENO integration.Convection-diffusion equations are fundamental equations in viscous fluid dy-namics and other applications. As usual, exact solutions are not available for most non-linear convection-diffusion equations, therefore the study of numerical methods is very important in theory and applications. The main challenge to the design of numerical schemes for solving convection-diffusion equations, especially the convection domi-nated or degenerate parabolic cases, is the presence of discontinuous or sharp transition layers. WENO schemes were introduced to approximate hyperbolic conservation laws and the first derivative convection terms in convection dominated convection-diffusion equations. The usual practice in solving a convection-diffusion equation by WENO schemes is to use the WENO procedure only to approximate first derivative terms and to use standard central difference to approximate second derivative diffusion terms. While this works well in most situations, it would lead to oscillations or even instabil-ities when applied to certain degenerate parabolic equations containing discontinuous solutions. We develop conservative high order finite difference WENO approximations to second derivatives, and use the porous medium equation as an example to discuss their application to nonlinear degenerate parabolic equations which may contain dis-continuous solutions. Both1D and2D nonlinear convection-diffusion equations are considered. Nonlinear degenerate parabolic equations have features similar to those of hyperbolic conservation laws, such as the possible existence of sharp fronts and finite speed of propagation of wave fronts. One possible approach is to apply the WENO procedure to two first derivatives rather than to the second derivative term directly, by introducing an auxiliary variable for the first derivative. However, the effective stencil and the computational cost are both relatively large. Therefore, we directly approxi-mate the second derivative term using a conservative flux difference. We first study the linear weights for various numbers of stencils, construct the sixth order and eighth or-der finite difference WENO schemes, and then analyze the accuracy of the schemes and the CFL stability conditions. Finally numerical examples demonstrating the accuracy and non-oscillatory performance of the sixth order and eighth order WENO schemes are provided.An important property of solutions to convection-diffusion equations is that they satisfy a strict maximum principle. This property is also desirable for numerical so-lutions, as otherwise the solutions may be physically meaningless, for example, in the radionuclide transport calculations. The numerical solutions of the porous medium equation computed by the finite difference WENO schemes may become negative near the sharp fronts, even though they are essentially non-oscillatory. Therefore, we gener-alize the maximum-principle-satisfying schemes for scalar conservation laws, propose a non-conventional high order finite volume WENO scheme, and extend the scheme to2D problems. The key idea is to introduce the concept of double cell averages, by applying the cell averaging operator twice, and then form a high order accurate WENO reconstruction based on these double cell averages. The resulting scheme can be shown to be genuinely high order accurate and satisfy the maximum principle under suitable CFL stability conditions. We test the fifth order finite volume WENO scheme, and observe the fifth order accuracy, non-oscillatory performance and the strict maximum principle.