Unit 2: Numerical Methods for Partial Differential Equations

2.1 Overview

• 2.1.1 Measurable Outcomes
• 2.1.2 Pre-requisite Material

2.2 Partial Differential Equations

• 2.2.1 Conservation Laws in Integral and Differential Form
• 2.2.2 One-Dimensional Burgers Equation
• 2.2.3 Convection
• 2.2.4 Characteristics for One-Dimensional Burgers Equation
• 2.2.5 Diffusion
• 2.2.6 Convection-Diffusion
• 2.2.7 Linear Elasticity

2.3 Introduction to Finite Difference Methods

• 2.3.1 Finite Difference Approximations
• 2.3.2 Finite Difference Methods
• 2.3.3 Finite Difference Method Applied to 1-D Convection
• 2.3.4 Forward Time-Backward Space FTBS

2.4 Analysis of Finite Difference Methods

• 2.4.1 Local Truncation Error for a Derivative Approximation
• 2.4.2 Truncation Error of Central Difference Approximation
• 2.4.3 Truncation Error for a PDE
• 2.4.4 Finite Difference Methods in Matrix Form
• 2.4.5 General Finite Difference Approximations
• 2.4.6 Boundary Conditions for Finite Differences

2.5 Introduction to Finite Volume Methods

• 2.5.1 Finite Volume Method in 1-D
• 2.5.2 Finite Volume Method Applied to 1-D Convection
• 2.5.3 Finite Volume Method in 2-D
• 2.5.4 Finite Volume Method for 2-D Convection on a Rectangular Mesh
• 2.5.5 Finite Volume Method for Nonlinear Systems

2.6 Upwinding and the CFL Condition

• 2.6.1 Upwinding
• 2.6.2 The CFL Condition

2.7 Eigenvalue Stability of Finite Difference Methods

• 2.7.1 Fourier Analysis of PDEs
• 2.7.2 Matrix Stability for Finite Difference Methods
• 2.7.3 Circulant Matrices
• 2.7.4 Stability Exercises

2.8 Method of Weighted Residuals

• 2.8.1 Functional Approximation of the Solution
• 2.8.2 The Collocation Method
• 2.8.3 The Method of Weighted Residuals
• 2.8.4 Galerkin Method with New Basis

2.9 Introduction to Finite Elements

• 2.9.1 Motivation
• 2.9.2 1-D Finite Element Mesh and Notation
• 2.9.3 1-D Linear Elements and the Nodal Basis
• 2.9.4 Weak Form of the Weighted Residual
• 2.9.5 Calculation of the Finite Element Weighted Residual
• 2.9.6 Calculation of the Stiffness Matrix

2.10 More on Finite Element Methods

• 2.10.1 Gaussian Quadrature
• 2.10.2 Boundary Conditions for Finite Elements

2.11 The Finite Element Method for Two-Dimensional Diffusion

• 2.11.1 Overview
• 2.11.2 Reference Element and Linear Elements
• 2.11.3 Differentiation using the Reference Element
• 2.11.4 Construction of the Stiffness Matrix
• 2.11.5 Integration in the Reference Element