Numerical solutions of partial differential equations.

Most of the phenomena in varying fields such as Space Science, Meteorology and Civil Engineering are governed by complex Partial Differential Equations (PDEs), which in most cases have no analytical solutions. With computers, numerical methods such as the Finite-Difference (FD) method have become invaluable tools in approximating solutions of real life problems. This has given us insight into these physically or geometrically complex problems. We shall seek to give a brief introduction to FD methods and general second-order PDEs. The FD method will be applied to the diffusion and advection equations, paying close attention to numerical stability and diffusiveness. We note that the advection equation can be solved with various flux assignment schemes. In this study we consider the Forward-Time Central Space (FTCS), Lax-Wendroff and the Forward Upwind Schemes. We further note that the latter gives the best approximation. However, it is highly diffusive and requires the use of so called flux-limiters such as the Van Leer, Minmod and Superbee to limit it's diffusiveness. After illustrating the workings of flux-limiters, we developed our own and compare it to the well-known limiters.