Approximation of parabolic PDEs with a discontinuous initial condition
Piet Hemker, Gregorii I. Shishkin · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 1993
Approximation of parabolic PDEs with a discontinuous initial condition P. W. HEMKER• and G.I. SHISHK!Nt Ahltnct -We CODlider a Dirichlet problem for a parabolic partial dilferential equation wilb a diacontmuous initial c:ooditioD.'The boundary CDllditioD II: t • 0 ia ~to have a discontinuity ol lhe .&rscJcind.Duo to the smgalarity of the solutiOll in the neighbourhood ol the discontinuity, tbe usual diacretizalion methods do not jield convcrgcace in the 1!"'-norm in the entire domain of dcfuUtion.ThereCore, In order to handle the singularity u adapted scheme is coastructccl.We use a specially fitted diffCl'Cllcc operator ea a zesu!ar rectangular grid Such a difference aehemc con-sea in lbc discrete !!"-norm on the wbolc unifonn grid Por a model problem, 11wnerical experiments with tbe c:lusica1 Biid the specially fitted scbcmea are compared 111d discussccl.~.Parlbolic PDE, discontinuous boundary axulilicm, finite difference mctbods, uniform ooavcr• gencc.Solutions of parabolic boundary value problems with discontinuous initial conditions are not smooth on their domain of definition.Therefore, difficulties arise when these problems are solved by numerical methods.As was shown, e.g. in [6,7) the solutions of difference equations which are constructed on regular rectangular grids using classical schemes do not converge in the i""-norm in the neighbourhood of the discontinuity in the boundary condition.Our aim is to construct a scheme which converges in the t"'-norm throughout the domain of definition.Different approaches can be used for constructing of such special schemes for problems with non-smooth solutions: (1) methods in which the singularity is split off and represented separately (e.g. by introducing special basis functions in the Fmite Element Method); (2) methods that use special, refined meshes in the neighbourhoods of sin• gularities; (3) fitted methods in which the coefficients of the difference equations are adapted to the singularities.A method combining the second and the third approach was proposed in [6,7].A second-order one-dimensional parabolic equation with a discontinuous boundary condition was studied; the highest derivative of the equation contained a small parameter e E (0, 1).When e -+ 0, the equation reduces to an equation with only a first-order derivative for the time-variable.A special difference scheme was constructed for this singularly perturbed boundary value problem.This scheme converges uniformly with respect to the small parameter in the i""-norm on the whole domain.Outside some neighbourhood of this discontinuity the classical difference scheme was used on a rectangular grid.In the neighbourhood of the discontinuity special parabolic variables asCWI, Amltcrdam, The Netherlands tinstitute ol Mathematics and Mechanics, !he Urals Brallcb of the IWuian Aaul.Sci., Elcateri11burg, R.uuia TIDa =earch wu supported in part by lbe Dutch RclCarcb OrganizatiOll NWO, vant No. 07-30-012.