Abstract
After discretizing the spatial derivatives in the governing PDE’s (such as the Navier-Stokes equations), we obtain a coupled system of nonlinear ODE’s in the form
These can be integrated in time using a time-marching method to obtain a time-accurate solution to an unsteady flow problem. For a steady flow problem, spatial discretization leads to a coupled system of nonlinear algebraic equations in the form
As a result of the nonlinearity of these equations, some sort of iterative method is required to obtain a solution. For example, one can consider the use of Newton’s method, which is widely used for nonlinear algebraic equations (see Section 6.10.3). This produces an iterative method in which a coupled system of linear algebraic equations must be solved at each iteration. These can be solved iteratively using relaxation methods, which will be discussed in Chapter 9, or directly using Gaussian elimination or some variation thereof.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lomax, H., Pulliam, T.H., Zingg, D.W. (2001). Time-Marching Methods for ODE’S. In: Fundamentals of Computational Fluid Dynamics. Scientific Computation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04654-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-662-04654-8_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07484-4
Online ISBN: 978-3-662-04654-8
eBook Packages: Springer Book Archive