Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- Flowchart of contents
- I Ordinary differential equations
- 1 Euler's method and beyond
- 2 Multistep methods
- 3 Runge–Kutta methods
- 4 Stiff equations
- 5 Geometric numerical integration
- 6 Error control
- 7 Nonlinear algebraic systems
- II The Poisson equation
- III Partial differential equations of evolution
- Appendix Bluffer's guide to useful mathematics
- Index
7 - Nonlinear algebraic systems
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- Flowchart of contents
- I Ordinary differential equations
- 1 Euler's method and beyond
- 2 Multistep methods
- 3 Runge–Kutta methods
- 4 Stiff equations
- 5 Geometric numerical integration
- 6 Error control
- 7 Nonlinear algebraic systems
- II The Poisson equation
- III Partial differential equations of evolution
- Appendix Bluffer's guide to useful mathematics
- Index
Summary
Functional iteration
From the point of view of a numerical mathematician, which we adopted in Chapters 1–5, the solution of ordinary differential equations is all about analysis – i.e. convergence, order, stability and an endless progression of theorems and proofs. The outlook of Chapter 6 parallels that of a software engineer, being concerned with the correct assembly of computational components and with choosing the step sequence dynamically. Computers, however, are engaged neither in analysis nor in algorithm design but in the real work concerned with solving ODEs, and this consists in the main of the computation of (mostly nonlinear) algebraic systems of equations.
Why not – and this is a legitimate question – use explicit methods, whether multistep or Runge–Kutta, thereby dispensing altogether with the need to calculate algebraic systems? The main reason is computational cost. This is obvious in the case of stiff equations, since, for explicit time-stepping methods, stability considerations restrict the step size to an extent that renders the scheme noncompetitive and downright ineffective. When stability questions are not at issue, it often makes very good sense to use explicit Runge–Kutta methods. The accepted wisdom is, however, that, as far as multistep methods are concerned, implicit methods should be used even for non-stiff equations since, as we will see in this chapter, the solution of the underlying algebraic systems can be approximated with relative ease.
Let us suppose that we wish to advance the (implicit) multistep method (2.8) by a single step.
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- Publisher: Cambridge University PressPrint publication year: 2008