New Numerical Analysis on Characteristic Type Methods for Nonlinear Parabolic Partial Differential Equations

The convection and diffusion play an important role in many physical phenomena. Insome applications, the convection may dominate in the fluid transport process and thediffusion is weak. In such a case, the model is often described by a parabolic equation (orsystem) with a strong hyperbolic feature. This festure should be considered in developingnumerical methods. The method of characteristic type, which is based on a hyperbolictracking, is especially effective for convection-dominated diffusion equations andnumerical simulations show that the method is stable for a large time step. However,existing theoretical analyses may not match such observations well since optimal errorestimates were always obtained under certain time stepsize restrictions. In this project,we intend to investigate further characteristic type methods for a large class ofnonlinear convection-dominated diffusion equations and to provide unconditional erroranalysis.