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Numerical approximations to interface curves for a porous media equation. (English) Zbl 0537.76065
The author investigates difference approximations to the initial value problem for the equation $$(*)\quad v_ t=(v^ m)_{xx}$$, $$t\in(0,\infty)$$, $$x\in R^ 1$$, $$(m>1)$$ with an initial value $$v(0,x)=v^ 0(x)$$, $$x\in R^ 1$$. Here v represents the density of an ideal gas flowing in a homogeneous porous medium. It is known that when $$v^ 0(x)$$ has compact support, a solution v(t,x) of (*) has also compact support for any $$t>0$$. Therefore a curve, which separates the region with $$v>0$$ from the region with $$v=0,$$ can be defined. This curve is called the interface $$\lambda$$ (t) and its two parts $$(j=1,2)$$ are governed by $\frac{d}{dt}\lambda_ j(t)=-(m/(m-1))\lim_{x\to \lambda_ j(t)}(v^{m-1})_ x(t,x).$ These curves and some of their critical points are to be approximated.
To construct a difference scheme, the authors set $$u=v^{m-1}$$ into (*) and split the resulting operator in two parts $$Pu=muu_{xx}$$, $$Hu=a(u_ x)^ 2$$ so that the equation is now $$u_ t=Pu+Hu$$, $$a=m/(m-1), u(0,x)=u^ 0(x)=(v^ 0(x))^{m-1}$$.
The described method starts with a difference scheme for the equation $$u_ t=Hu$$ and using some known properties of this equation a difference scheme for the equation $$u_ t=Pu$$ is constructed. Stability and convergence properties are formulated and proved, one numerical example is given. The details of the method cannot be described here: The method seems to give not only approximations of the solution v(x,t) but also approximations of the interface curves and further important information.
Reviewer: J.Gregor

MSC:
 76S05 Flows in porous media; filtration; seepage 76M99 Basic methods in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs