On 2/12/2012 3:28 PM, Gib Bogle wrote: > I'm showing my ignorance by asking this question, but here goes... > > It's possible to represent a discretised PDE by a system of ODEs. For > example, the simple 1D diffusion equation > > dC/dt = k.d2C/dx2 (where derivatives are partial) > > can be represented on a grid of points with spacing h by > > dC(i)/dt = K.(C(i-1) -2.C(i) + C(i+1))/h^2 > > With suitable treatment of initial and boundary conditions, this ODE > system could be solved with one of several methods. > > Is there a significant difficulty with this approach?
It is more a question of efficiency rather than difficulty. The spatial discretization has error O(h^2); if mixed boundary conditions apply and are not discretized properly, the error can degenerate to O(h). In such circumstances, using a fourth order ODE solver such as RK4 is overkill. If the equation being solved has a source term that depends on C, the problem is nonlinear and iterations become necessary. In such cases, efficiency becomes important.
For these reason, it is more common to use first order integration schemes (Forward and Backward Euler) or second order (Midpoint) time-differencing than higher order schemes such as RK4.