From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci) Subject: Re: Seeking help on Richards equation Date: 14 May 1999 18:52:31 GMT Newsgroups: sci.math.num-analysis Keywords: solving PDEs with discontinuous coefficients [35R05] In article , hvaisane@joyx.DELETE_THISjoensuu.fi (H. Väisänen) writes: |> I have not succeeded in solving this problem (I am using ddaspk) and help |> is appreciated. |> |> I should model the water content of soil with the Richards equation |> |> \begin{equation} |> \frac{\partial \theta(t,x)}{\partial t} |> = \frac{\partial}{\partial x} |> \left( D(\theta) \frac{\partial \theta}{\partial x} \right) |> + \frac{\partial K(\theta)}{\partial x} |> \end{equation} |> with |> $D(\theta)$ and $K(\theta)$ are known functions |> but $D(\theta)$ is not continuous. you cannot solve PDE's with discontinuous coefficients using the method of lines with a standard integrator, since this method assumes evrything being smooth (even smooth to high order!) you must solve the problem as a free boundary problem for identifying the boundary where the coefficient D jumps and solve the complete problem piecewise. also there occur continuity conditions for the first derivative along and across this boundary which are to be used for fitting the pieces together. unfortunately, i canot help you directly because i don't know a code for doing this. a literature search revealed very little, not to speak about ready solutions. see below. you may also make a literature search yourself in math class 35r05 (PDE's with discontinuous coefficients). that the initial and boundary values are given as discrete data should be no problem, you can approximate them beforehand by splines. an idea might be to use the method of lines combined with an interpolation process to identify the boundary using an explicit code for time integration suitable for mildly stiff equations as rkc.f of verwer and sommeijer, which is available through netlib/ode, because this gives you the opportunity directly to control where the jumps occur. you then must reset the spatial grid such that the point of discontinuity becomes a grid _and_ boundary point, recompute the current u as a new initial value (again by an interpolating spline), do a restart and so on. just an idea, but needs careful analysis whether this is mathematically correct. good luck peter this is from "zentralblatt fuer mathematik" 758.65062 Wang, Wenqia A numerical method for a three-dimensional parabolic equation with discontinuous coefficient and its numerical analysis. (Chinese. English summary) [J] J. Shandong Univ., Nat. Sci. Ed. 27, No.2, 149-157 (1992). An integrating interpolation scheme is set up to solve a three-dimensional parabolic equation with discontinuous coefficient. The approximation error is given, and stability and convergence theorems are proved. [ Hou Zongyi (Shanghai) ] Schlüsselwörter: error bound; integrating interpolation scheme ; three-dimensional parabolic equation; discontinuous coefficient; stability; convergence Johnston, Peter R. Diffusion in composite media: Solution with simple eigenvalues and eigenfunctions. (English) [J] Math. Comput. Modelling 15, No.10, 115-123 (1991). A method was presented to solve the governing equations of the form \par $(1/x\sp s)(\partial/\partial x)(x\sp s k(x)\partial T/\partial x) =\rho(x)c\sb p(x)\partial T/\partial t$ for transition temperature distribution in composite media. The method is based on the technique of Sturm-Liouville finite integral transforms and uses the eigenvalues and eigenfunctions which arise naturally from the geometry of the problem. The method was originally proposed by {\it D. D. Do} [Chem. Eng. Sci. 39, 1519 ff. (1984)].\par The governing equations must be integrated twice to remove the necessity of differentiating the discontinuous coefficient functions. There is no difficulty to find all eigenvalues. As examples the temperature profiles were calculated and plotted for a three layered slab, a two layered cylinder, and a two layered sphere. [ V.Burjan (Praha) ] Schlüsselwörter: transition temperature distribution ; Sturm-Liouville finite integral transforms; three layered slab; two layered cylinder; two layered sphere 734.35162 Bai, Donghua; Tan, Qijian Local solutions of a free boundary problem. (Chinese. English summary) [J] J. Sichuan Univ., Nat. Sci. Ed. 28, No.2, 151-157 (1991). We establish a local existence theorem for a one dimensional free boundary problem suggested by a nonlinear parabolic equation with discontinuous coefficient. The equation simulates the flow of two slightly compressible and inviscid fluids in a vertical porous medium. Schlüsselwörter: local existence; free boundary problem ; parabolic equation; discontinuous coefficient Donato, Santi A normal derivative problem for a class of parabolic equations of several variables with discontinuous coefficients. (Italian. English summary) [J] Rend. Mat. Appl., VII. Ser. 7, No.2, 255-272 (1987). Soit $\Omega$ un ouvert borne de $R\sp m$ $(m>2)$, $T>0$. Dans $\Omega \times [0,T]=Q$ on considere le probleme $$ (*)\ {\cal L}u=\sum a\sb{ik}(x,t)\partial\sp 2u/(\partial x\sb i \partial x\sb k)+\sum b\sb i(x,t)\partial u/\partial x\sb i+c(x,t)u-u\sb t =f(x,t) $$ dans Q, $u(x,0)=0$ dnas $\Omega$ et $\partial u/\partial n+\alpha (x,t)u=0$ sur $\partial \Omega \times [0,T]$. On suppose les $a\sb{ik}$ mesurables dans Q, et verifiant l'estimation $$ \sum a\sb{ik}+1/(\sum a\sb{ii}+1)\le 1/(m+\epsilon)\ avec\ 0<\epsilon \le 1 $$ presque partout dans Q. Les $b\sb i\in L\sp{p,q}(Q)$ avec $m\le p\le \infty$ et $q=2p/(p-m)$ (si $p=m$ on a une autre evaluation). \par Le coefficient $c\in L\sp{r,s}$ avec r[2,3] si $m=3$, et, $r\in (2,4]$ si $m=4$ et enfin $m/22$. [ G.Gussi ] Zitate: Zbl.505.35043; Zbl.262.35026; Zbl.609.35044; Zbl.156.340 Schlüsselwörter: oblique derivative problem ; measurable coefficients; unique solvability; $L\sp 2$-estimates for the Hessian