Re: explicit versus implicit method for the heat equation

Helmut Jarausch wrote:
Chris wrote:


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?


Yes and No. If you have 'rough' initial values (at time "t=0") AND
you want to follow the initial, short transient phase, you're right,
you have to take small time steps. (Note, the same situation
can occur for "positive time" if you have time dependent source
terms our the equation is non-linear)

Note, even if the solution is (looks) smooth the time step of an
explicit scheme is restricted by the highest possible frequencies
even if these aren't present in the current solution. If the time
step is too large, even tiny high frequency parts (e.g. due to rounding
errors) blow up immediately.
On the contrary, with an implicit scheme which is A-stable (or L-stable
if the eigenvalues of the operator in space are real) high frequency
components will be damped out immediately for all time steps.


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)?


No, see above

Helmut Jarausch

Lehrstuhl fuer Numerische Mathematik
RWTH - Aachen University
D 52056 Aachen, Germany


Dear Helmut,

thanks for your answer, it is quite in line with what I already suspected. However, only one thing is not yet fully clear to me. If I am correct, for the specific case of the diffusion equation, the global error of both the forward and backward euler scheme is

e=O( C1 * d^2u/dt^2 * dt - C2 * d^4u/dx^4 * dx^2 )

Using the PDE itself, this can be rewritten as

e=O( (C1*dt+C2*dx^2) * d^4u/dx^4).

As far as I know, the constants C1 and C2 are now independent of the solution itself (but they depend on, e.g., the end time of the integration).

If the above is correct, it is in line with what you are saying:

1)if we are integrating over _short times_, C1 will be of the same magnitude of C2, and dt must be of the same magnitude as dx^2.

2)if we want to integrate over _longer times_ (to find, e.g., a state near a steady-state), then I can understand that the error due to the time integration becomes smaller and smaller, until we only have the spatial discretisation error left in the steady-state. My intuitive explanation for this is that if the scheme is stable, then the errors also go to zero as time goes to infinity (or is this too simplistic?). Hence in that case the constant C1 would probably be substantially smaller than C2, and we may take larger time steps (with the implicit scheme).


The only thing that I do not understand is why you are speaking about "rough" initial values: the roughness (or derivatives) appear in both the time and space discretization (for my specific model problem), and according to the formula above they should hence not matter (right?).
So the roughness or derivatives determine the actual size of the error e, but not how dt whould be chosen compared to dx.


Do you agree with my reformulation of your answer? Or is there still something wrong with my reasoning?


Thanks in any case for your excellent help,

Chris
.

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