Numerical approximations to interface curves for a porous media equation.

*(English)*Zbl 0537.76065The 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.

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 |