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2.1 Overview

2.1.1 Measurable outcomes

In the last module we studied numerical methods for ODEs. In this module, we will will develop numerical methods for partial differential equations (PDEs), which arise in many different physical systems.

Specifically, students successfully completing this module will be able to:

  • Measurable Outcome 2.1: Identify whether a PDE is in the form of a conservation law, describe the characteristic of a conservation law and how the solution behaves along the characteristic.

  • Measurable Outcome 2.2: Qualitatively describe the solution to simple PDEs: convection equation, diffusion equation, convection-diffusion equation, Burgers equation.

  • Measurable Outcome 2.3: Implement a finite difference or finite volume discretization to solve a representative PDE (or set of PDEs) from an engineering application.

  • Measurable Outcome 2.4: Describe finite volume discretization of two-dimensional convection on an unstructured mesh.

  • Measurable Outcome 2.5: Define the physical domain of dependence for a problem.

  • Measurable Outcome 2.6: Define and determine the numerical domain of dependence for a discretization.

  • Measurable Outcome 2.7: Explain the CFL condition and determine the timestep constraints resulting from the CFL condition.

  • Measurable Outcome 2.8: Determine the local truncation error for a finite difference approximation of a PDE using a Taylor series analysis.

  • Measurable Outcome 2.9: Explain the difference between a centered and a one-sided (e.g., upwind) discretization.

  • Measurable Outcome 2.10: Define eigenvalue stability.

  • Measurable Outcome 2.11: Perform an eigenvalue stability analysis of a finite difference approximation of a PDE using either Von Neumann analysis or a semi-discrete (method of lines) analysis.

  • Measurable Outcome 2.12: Describe how the method of weighted residuals can be used to calculate an approximate solution to a PDE.

  • Measurable Outcome 2.13: Describe the differences between the method of weighted residuals, the collocation method, and the least-squares method for approximating a PDE.

  • Measurable Outcome 2.14: Describe the Galerkin method of weighted residuals.

  • Measurable Outcome 2.15: Describe the choice of approximate solutions (i.e., the test functions or interpolants) used in the finite element method.

  • Measurable Outcome 2.16: Give examples of a basis for the approximate solutions, in particular, including a nodal basis for at least linear and quadratic solutions.

  • Measurable Outcome 2.17: Describe how integrals are performed using a reference element.

  • Measurable Outcome 2.18: Explain how Gaussian quadrature rules are derived. Describe how Gaussian quadrature is used to approximate an integral in the reference element.

  • Measurable Outcome 2.19: Explain how Dirichlet and Neumann boundary conditions are implemented for Laplace's equation, discretized by the finite element method.

  • Measurable Outcome 2.20: Describe how the finite element method discretization results in a system of discrete equations and, for linear problems, gives rise to the stiffness matrix. Describe the meaning of the entries (rows and columns) of the stiffness matrix and of the right-hand side vector for linear problems.