Differential Equations in Numerical Modeling

COURSE DESCRIPTION

The purpose of the course is to present numerical methods for solving differential equations arising from modeling real-world problems. The underlying mathematics of the numerical methods are an integral part of the course. Each chapter starts with the derivation of a model for a physical problem in terms of differential equations. Then a numerical method for solution is introduced and analyzed.

Prerequisites

Matrix Algebra, Advanced Calculus, Elements of Ordinary Differential Equations

COURSE GOALS

On completion of the course, students will

  • Understand the basics of mathematical modeling. From physical problem to equation
  • Know the properties of numerical methods to make informed choices for solution
  • Be able to do numerical simulation in a software environment such as Matlab.
  • Be proficient in the underlying mathematics of numerical methods
COURSE CONTENTS
  1. Ordinary differential equations. Embedded Runge-Kutta methods. Parameter estimation in ODE (1 week).
  2. Finite difference method. Potential and diffusion (1 week).
  3. Finite volume method. Examples in Computational Fluid Dynamics (CFD) (2 weeks).
  4. The finite element method for elasticity (2 weeks).
  5. Discontinuous Galerkin method. Scalar conservation laws (2 weeks).

Bibliography

  1. Cooper, Jeffery M. (2000): Introduction to Partial Differential Equations with MatLab, Boston: Birkhäuser.
  2. Van Groesen, E.; Molenar, Jaap (2007). Continuum Modeling in the Physical Sciences, Philadelphia: SIAM.
  3. LeVeque, Randall J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations, Steady-State and Time-Dependent Problems, Philadelphia: SIAM.
  4. Mattheij, R.M.M.; Rienstra, S.W.; ten Thije Boonkkamp, J.H.M. (2005). Partial Differential Equations; Modeling, Analysis, Computation, Philadelphia: SIAM.
  5. Quarteroni, Alfio; Sacco, Riccardo; Saleri, Fausto (2000). Numerical Mathematics, New York: Springer.
  6. Stoer, Josef; Bulirsch, R. (2002). Introduction to Numerical Analysis, Third Edition, New York: Springer-Verlag.

Support Sessions

2 hours per week with a teaching assistant

Grading

Homework (40%), midterm exam (30%), final project (30%)