**Course No.****：**S070102ZJ002

**Course Category****：**Professional Basic Course

**Period/Credits****：**40/2

**Prerequisites****：**Advanced Mathematics, Linear Algebra, Equations of Mathematical Physics.

**Aims & Requirements****：**

This course is a speciality foundation course for Master degree candidates majoring in computational mathematics. It can also be taken as an elective course for Master degree candidates majoring in other mathematical subjects. The course introduces difference methods and finite volume methods for numerically solving partial differential equations of parabolic and hyperbolic types, and their applications to the computation of fluid flow. Participants of this course are expected to command some usual methods of numerically solving partial differential equations, to lay the foundation for further study of computation of flows.

**Primary Coverage****：**

Chapter 1 Theoretical Foundation of Finite Difference Methods

Methods of constructing difference schemes; General requirements on difference schemes; Lax’s equivalence theorem; von Neumann’s stability analysis for difference schemes; modified equations of difference schemes.

Chapter 2 Difference Methods for Linear equations of Parabolic Type

Explicit schemes for diffusion equations; implicit schemes for diffusion equations; methods of lines; ADI methods for multi-dimensional equations of parabolic type; dimension splitting method for multi-dimensional equations of parabolic type; difference methods and mesh Reynolds number for Burgers equation.

Chapter 3 Numerical Methods for 1-dimensional Linear Equations of Hyperbolic Type

Characteristics and Riemann problem of linear hyperbolic systems; finite volume methods for conservation laws; Lax-Friedrichs schemes, Lax-Wendroff schemes, difference schemes of characteristic line methods; upwind schemes, CIR schemes, and Godunov methods for hyperbolic equations; second order Godunov schemes, total variation and limiter function; high-resolution (TVD) wave propagation schemes for hyperbolic equations (systems) and that with varying coefficients.

Chapter 4 Numerical methods for 1-dimensional non-linear hyperbolic conservation law

Discontinuous solution, weak solution, and entropy condition for non-linear hyperbolic conservation law; Solution of Riemann problem and Godunov scheme for scalar conservation law; entropy modification, numerical viscosity, Osher scheme, and high-resolution wave propagation scheme; conservational methods and Lax-Wendroff theorem, discrete entropy condition, non-linear stability and convergence; Godunov discontinuous decomposition method and Godunov scheme for classical conservation law equations systems; MUSCL scheme for conservation law equations systems.

Chapter 5 High-resolution schemes for multi-dimensional hyperbolic conservation law

Hyperbolicity of multi-dimensional equations systems; Lax-Wendroff methods, semi-discrete methods, dimension splitting methods; LW method, Godunov scheme, direction and angle upwind schemes of scalar equations; high-resolution schemes of multi-dimensional scalar equations; high-resolution schemes of multi-dimensional equations systems.

Chapter 6 Other high-resolution methods for hyperbolic conservation law

ENO and WENO scheme; discontinuous Galerkin method; high-resolution compact difference schemes.

**References****：**

[1] R. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.

[2] C.A.J. Fletcher, Computational Techniques for Fluid Dynamics 1, (second edition), Spinger-Verlag, 1991.

[3] R. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM publishing, 2007.

**Author****：**Li Yuan (Academy of Mathematics and Systems Science)

**Date****：**June, 2010