Numerical solutions

Description

This module starts by reviewing the Newton solver in 1D and 2D. Linear and non-linear equation solving methods are briefly touched upon before the basics of solving an Ordinary Differential Equation (implicit and explicit Euler methods) are discussed. The method of lines for solving Differential Algebraic Equations (DAEs) is explained. Discretization (finite difference, finite element and finite volume), orders of approximation and number of points are all discussed. Round-off and truncation errors are explained. Numerical dispersion, high-index problems and scaling are also covered.


Module preview


Content

  • Algebraic and non-linear equation solving
  • Solving of ordinary differential equations
  • Method of lines
  • Discretization
  • Order of approximation
  • Finite difference, element and volume methods
  • Errors in discretization
  • Selection of method, order and number of points
  • Initial and boundary conditions (number, place and type)
  • Numerical dispersion/diffusion
  • High-index and scaling

Learning outcomes

At the end of this module, students should have a grasp of what numerical techniques are used in solving DAE systems, how to select discretization method, order and number of points. They should have a basic understanding of numerical artifacts such as high-index problems, scaling etc.


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