Solution of parabolic partial differential equations by a double collocation method

✍ Scribed by John Villadsen; Jan P. Sørensen

• Publisher: Elsevier Science
• Year: 1969
• Tongue: English
• Weight: 1023 KB
• Volume: 24
• Category: Article
• ISSN: 0009-2509
• DOI: 10.1016/0009-2509(69)85054-2

✦ Synopsis

A method for solution of parabohc PDE by means of mterpolahon of the Merentlal operators m two &menslons 1s described The method IS developed on the basis of previously pubhshed methods for solution of ordmary &fferentiaI equations by orthogonal collocation It 1s shown to be highly econonucal and very stable m comparison with the conventional Crank-Nlcolson or wth exphclt methods such as the Runge-Kutta 4th order method The hnear heat equation IS used to dlustrate the pnnclple of the method and to discuss Its convergence propertles

THE MOST commonly used method for the numerical solution of parabolic parttal dtfferenteal equations is the finite difference method The explicit methods, which are the simplest from a computational standpoint are usually avotded, due to mstabthty problems, that make it necessary to use very small mcrements m the direction of the "ttme" variable t The two-level imphcit methods approxunate the second order spatial derivative 0,, by a weighted average between column J and column (J+ 1) m the dtfference scheme If the weight factors of column J and column (J + 1) are respectively 1 -h and A the following approximation results ~1~~ = &h4-lJ+l -2h+, + ei+l,+li + (Ax)~ *[6-1J-26J+~8+1Jl (I)