Numerical PDE Techniques for Scientists and Engineers

by: Prof. Dinshaw S. Balsara

Chapter 2: Finite Difference Approximations

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Abstract

The first chapter introduced us to several of the different kinds of partial differential equations (PDEs) that govern the evolution of physical systems. Our purpose in this book is to study numerical techniques for the solution of these equations. The computer only solves a discrete approximation for the actual PDE. It is not always guaranteed that the numerical approximation saliently converges to the solution of the PDE. Obtaining such guarantees is the task that we undertake in this chapter.

Any numerical method should carry some guarantees that it will converge to the physical solution. Even for the very simple case of a scalar, linear PDE, it is not guaranteed that any numerical method that one might devise will converge to the physical solution. In this chapter, we take our first stab at obtaining such guarantees. For parabolic equations such guarantees will indeed be obtained in this chapter. For hyperbolic problems such a study will spill over to the next chapter. The solution methods for all the PDEs mentioned in this chapter are described in the mathematical literature as initial boundary value problems. They consist of specifying a set of initial conditions in the domain of interest along with self-consistent boundary conditions at the boundary of the domain and then evolving the solution of the given PDE in time. In this chapter we also begin a study of boundary conditions for PDEs.

Outline of Chapter

• 2.1) Introduction
• 2.2) Meshes and Discretization on a Mesh (Slides 1, Slides 2)
• 2.3) Taylor Series and Accuracy of Discretizations
• 2.4) Finite Difference Approximations and Their Consistency
• 2.5) The Stability of Finite Difference Approximations
• 2.6) von Neumann Stability Analysis for Linear Parabolic Equations
  • 2.6.1) Stability Analysis for Time-explicit Linear Parabolic Equations
  • 2.6.2) Stability Analysis for Time-implicit and Semi-implicit Linear Parabolic Equations
  • 2.6.3) Stability Analysis for the Time-implicit TR-BDF2 Method
  • 2.6.4) Boundary Conditions for Parabolic Equations
  • 2.6.5) Introduction to Matrix Methods for Parabolic Equations
• 2.7) von Neumann Stability Analysis of Linear Hyperbolic Equations
  • 2.7.1) Forward Euler Scheme (Never Used)
  • 2.7.2) Lax-Friedrichs Scheme
  • 2.7.3) The Lax-Wendroff Scheme
  • 2.7.4) The Two-stage Runge-Kutta Scheme
  • 2.7.5) First Order Accurate Upwind Scheme (i.e. Donor Cell Scheme)
  • 2.7.6) Section Summary for Hyperbolic Equations

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