Partial differential equations are of fundamental importance
in modeling many applications in science and technology.
Since in general it is difficult to find analytical solutions
for real-world problems, finding approximate solutions is necessary.
Finite difference methods are basic numerical methods for solving partial differential equations. In this course, we will only consider numerical
methods for mathematical models described by linear and nonlinear partial differential equations; the discussion of numerical methods for mathematical models governed by linear and nonlinear systems of partial differential equations will be considered in the next semester. As a whole, topics to be covered include:

Part I: Boundary Value Problems
1. Finite difference approximations,
2. Finite difference methods for elliptic equations,
3. Iterative solvers,
Part II: Time-Dependent Problems
4. Numerical ordinary differential equations
5. Stability and convergence
6. Diffusion equations
7. Hyperbolic equations