M371-M372: Numerical solution of time-dependent differential equations in applications

Goals and objectives of the course

I will offer a new two-semester course on the numerical integration of time-dependent differential equations (DEs) with a special emphasis on applications. There is enough material for a two-semester sequence of courses. The second semester will be at a higher level than the first one, and it will complement it well. The second semester can be considered as being a preparatory step for doing research work on the numerical solution of DEs whereas the first semester will be at a lower level. In these courses I will emphasize on time-dependent DEs arising in applications, especially in engineering. I would like each student to solve an application of his/her own interest. The project will be computer-oriented and will make use of the current available software, such as Fortran or Matlab codes. Each project will be the subject of a presentation in class to develop oral skills. It will also be the subject of a short report to be written. Both the report and the oral presentation will count for the final grade. The numerical solution of time-dependent DEs is a sufficiently broad topic to attract many engineering students. This course will cover the development, the mathematical analysis, and the use of practical algorithms for the numerical solution of time-dependent DEs. While initially offered as M371-M372, the longer run plan is to have a sequence numbered like M177-M277, as suggested by Ken Atkinson.

Differential equations in applications

Here is a tentative list of some possible applications to be discussed during the courses depending on the interests of the different departments and also of the students:

mechanical systems and multibody dynamics
electrical circuits
molecular dynamics
atmospheric chemistry
chemical reaction systems
convection-diffusion-reaction problems
optimal control problems
prescribed trajectory path control problems
structural dynamics

Classes of differential equations and related issues

We will treat in details the following classes of DEs:

ordinary differential equations (ODEs)
differential-algebraic equations (DAEs)

We will consider other classes of differential equations among the following ones depending on the interests raised and on the applications to be treated. We will give an emphasis on the classes of DEs with most interests in terms of applications and just a short introduction or no treatment at all on the ones with least interest. A detailed treatment will be given only in the more advanced course:

time-dependent partial differential equations (PDEs)
stochastic differential equations (SDEs)
delay-differential equations (DDEs)
differential inequalities (DIs)
discontinuous differential equations
differential inclusions

Some issues that will be discussed and which require different techniques for the diverse kinds of equations:

initial/boundary value problems (IVP/BVP)
autonomous/nonautonomous problems
definition of a solution
existence/uniqueness of a solution
linearity/nonlinearity

Some more advanced topics which could be treated only in the more advanced course:

stability of linear/nonlinear systems (Routh-Hurwitz criterion, Lyapunov functions)
differentiability/sensitivity of solutions with respect to parameters/initial values (Wronskian matrices, variational equations, the resolvent, the Groebner-Alekseev formula)
Sturm-Liouville eigenvalue problems
optimal control problems (the Pontryagin maximum principle)
parameter estimation

Certain of the following topics in variational calculus could be treated in relation with applications in mechanics:

first/second variations
geodesics
Euler-Lagrange equations
Lagrangian mechanics
Hamiltonian mechanics (symplectic applications)

Numerical integration of differential equations

We will consider diverse classes of methods and techniques to integrate numerically the various classes of DEs. We will treat in details the following class of methods in the less advanced course:

Taylor series
explicit/implicit Runge-Kutta (RK) methods

We will also treat some of the following classes of methods in the more advanced course:

explicit/implicit multistep methods (Adams/BDF methods)
collocation methods
Rosenbrock methods
extrapolation methods

The following topics will be considered:

the Euler method and its convergence
construction of higher order methods
existence/uniqueness of the numerical solution
local error analysis
order conditions
stability (see more details below)
global error analysis: consistency + stability => convergence
convergence order
nonstiff/stiff equations

Some advanced topics in numerical integration which could be treated in the more advanced course:

Butcher-tree analysis for the local error of Runge-Kutta methods
the symplifying assumptions of Runge-Kutta coefficients
use of the structure of the DEs to construct tailored methods
error constants
asymptotic expansion of global error
adjoint, symmetric methods
symplectic methods for Hamiltonian systems
volume-preserving flows and numerical methods
method of lines for time-dependent PDEs (leading to large-scale ODEs)
the Courant-Friedrich-Levy (CFL) conditions
single/multiple shooting for BVPs

Some of these advanced topics could be treated from a numerical perspective in the more advanced course:

critical points
periodic solutions
perturbation calculus
bifurcations point (e.g., Hopf bifurcations)
strange attractors
Poincare sections
stable/instable/central manifolds
homoclinic/heteroclinic orbits

Stability of numerical integration methods

In relation with numerical stability for linear ODEs we will treat the following topics in the less advanced course:

stability regions
A-stability

Some advanced topics in stability which could be treated in the more advanced course:

Pade approximations to the exponential function
order stars theory and the Ehle conjecture
logaritmic norms of matrices
B-stability
Dahlquist's barriers for multistep methods
preservation of invariants
backward error analysis
shadowing of numerical solutions

Implementation issues and software

Some issues are very important when writing software for differential equations and will be treated during the two semesters according to the main interests

solution of nonlinear systems for implicit methods (Newton-type methods, iterative solvers for linear systems)
error estimation (Richardson extrapolation, embedded methods)
stepsize control for variable stepsizes
order selection for multistep methods
dense output
event location
codes performance

An overview and use of current available software will be done during the courses along the semesters with as much as possible hands-on exercises.

Prerequisites

We require good programming skills, especially in Fortran or Matlab, but Pascal, C, or C++ are also acceptable. However, programming assistance wil be given only in Fortran and Matlab. We also require good knowledge of the theory of differential equations, of vector calculus, of linear algebra, and of numerical analysis in general (M170-M171 for example, at least M72).

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