Numerical Differential Equations

computer science
mathematics

elective courses

Remote learning

The course is devoted to a construction, analysis and implementation of fundamental methods for numerical solution of initial and boundary value problems for ordinary differential equations, and for boundary and initial-boundary value problems for basic type of partial differential equations: elliptic, parabolic and hyperbolic.

Ordinary differential equations with initial values. Multistep methods and Runge-Kutta methods and their analysis: convergence and stability, order of convergence, stiffness. Boundary value problems for these equations discretized by finite difference methods (FDMs) and finite element methods (FEMs).

Boundary value problems for linear elliptic equations of second order. Discretizations by FDMs and FEMs. Model problem for multidimensional Poison equation. A stability and convergence of FDMs and Galerkin methods (FEMs). Properties of discrete problems and their implementations.

Initial boundary value problems for linear and nonlinear parabolic equations. Explicit and implicate schemes, including Cranck-Nicolson one. Discretizations by FDM with respect time variable and by Galerkin (FEM) with respect space vanables. A convergence and stability theorem of these methods for linear equations. An implementation.

Initial and initial-boundary value problems for hyperbolic equations of first and second order. A discretization by FDM and FEM. A stability and order of convergence of these methods and their implementation.

1. D. Braess, Finite elements, Cambridge (2001)

1. A student knows basic numerical methods for solving ordinary differential equations with the initial value.

2. A student knows numerical methods for solving partial differential equations based on Finite Difference and Finite Element methods.

3. A student is able to select a right method with required properties of solving a given differential problem. He can analyse a method and implement it.

An oral exam