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
Finite difference approximations
Finite difference methods for elliptic equations
Immersed interface method for elliptic equations
Adaptive moving mesh method

Part II: Time-Dependent Problems
Numerical ordinary differential equations
Diffusion equations
Hyperbolic equations
Stability and convergence (Energy method and von Neumann analysis)

Part III: Applications
Shock waves and nonlinear conservation laws
Divergence-free constraint incompressible flows