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In my partial differential equations course, given an equation on $[0,T]\times\mathbb{R}^n$:

$\partial_t u(x,t) + b(x,t)\partial_x u(x,t) + a(x,t)\partial_{xx}u(x,t) + f(x,t) = 0$

$v(x,T) = g(x) \forall x\in\mathbb(R)^n$

We can define a finite-difference scheme $u_j^k, j\in\mathbb{Z}, k=0:N$ where $u_j^k$ is an approximation of $u(j\Delta x, k\Delta t)$ where $\Delta t = T/N$ a time step.

We say that the scheme is simply monotone if $\forall j, u_j ^{k-1}$ is an increasing function of $u_j^k, u_{j-1}^k, u_{j+1}^k$ and $f_j^k$.

In the course, we bother searching conditions on the time / space steps so that the scheme is monotone, but I do no understand the importance of such schemes.

Thank you very much,

Tom Mike
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1 Answers1

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There are several things which can go wrong.

1) Even for the simple heat equation the explicit forward difference scheme is only conditionally stable, and the stability condition is exactly the condition that guarantees monotonicity of the scheme.

2) It gets worse if the advection term $b$ dominates. Then good numerical approximations are obtained by upwind schemes; these again are monotone.

There is a discrete maximum principle for monotone schemes which guarantees that the "saw-tooth" instabilities do not grow and, more generally, the numerical solution has similar global properties as the analytic solution. There are more good reasons to use monotone schemes; just ask if more elaborated answer is required.

To really get the motivation for using monotone schemes, I'd suggest to experiment with schemes which are not monotone for the heat equation.

uvs
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