COURSE AIMS AND OBJECTIVES: The aim of the course is to introduce classical and modern methods of solving of linear and nonlinear partial differential equations. Finite difference method.
COURSE DESCRIPTION AND SYLLABUS:
1. Finite difference method. Application of the method to the Cauchz problem for linear parabolic and hzperbolic equation in one space dimension. Introduction of the notion of transport and diffusion and their numerical approximation. Consistence, stability and convergence of the method. Lax-Richtmyer equivalence theorem. CFL condition, region of influence for hyperbolic equation. Von Neumann stability analysis. Overview of basic schemes and their stability. Applications: testing of basic schemes in any programming language.
2. Finite difference method for nonlinear conservation laws. Hyperbolic conservation laws. Scalar equation: characteristics, shocks, Rankine-Hugoniot condition, viscous solution and entropz condition (Kružkov entropz condition). Discretization: conservative schemes, monotone schemes, TVD schemes. Applications: LeVeque software package CLawpack.
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