Time-Marching Methods for ODE’S

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

$$ \frac{{d\vec u}}{{dt}} = \vec F\left( {\vec u,t} \right) $$

(6.1)

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

$$ \vec F\left( {\vec u} \right) = 0 $$

(6.2)

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.