explicit versus implicit method for the heat equation


hi all,

Many people are solving the one-dimensional heat (diffusion) equation using central finite differences in space and forward Euler in time. Then, the time step dt is restricted by the well-known stability condition dt<dx^2/2/D.

To be able to take larger time steps, some people therefore use the implicit backward Euler scheme. Then, the time step can indeed be chosen larger than in the explicit case.

However, as both time integration methods are first order methods, and the error of the full numerical scheme (including spatial and temporal discretisation errors) is

e=O(C1*dt+C2*dx^2),

I am wondering if putting much effort in using the backward Euler scheme (which requires to solve a linear system) has much sense. Isn't the time step dt in this case also restricted to a condition dt<O(dx^2) due to accuracy arguments?

Or more generally stated: does it only make sense to use implicit time integration schemes to alleviate the stability restriction if the order of those schemes is at least 2 (still for the 1D diffusion equation of course)?


Any comments on my simplistic vision of numerical algebra are welcome!

Chris
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