Numerical applications of reflection to partial differential equations

Author:
Arthur David Snider

Journal:
Math. Comp. **30** (1976), 220-240

MSC:
Primary 65N99; Secondary 30A28, 65E05

DOI:
https://doi.org/10.1090/S0025-5718-1976-0443380-8

MathSciNet review:
0443380

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Abstract | References | Similar Articles | Additional Information

Abstract: Recent papers have reported results on the numerical solution of nonlinear free boundary problems wherein a conformal transformation (which must be determined) maps the unknown flow region onto a known domain; the boundary conditions are handled by the method of steepest descent. The present paper discusses the use of the reflection property of solutions of elliptic equations to determine these boundary conditions. The procedure is applied to the vena contracta models, and it is seen that it converges about ten times faster than the steepest-descent method.

**[1]**Hans Lewy,*On the reflection laws of second order differential equations in two independent variables*, Bull. Amer. Math. Soc.**65**(1959), 37–58. MR**0104048**, https://doi.org/10.1090/S0002-9904-1959-10270-6**[2]**Vladimir Filippenko,*On the reflection of harmonic functions and of solutions of the wave equation*, Pacific J. Math.**14**(1964), 883–893. MR**0170105****[3]**P. R. Garabedian,*Partial differential equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR**0162045****[4]**E. BLOCH,*A Finite Difference Method for the Solution of Free Boundary Problems*, AEC Computing and Applied Mathematics Center, Courant Inst. Math. Sci., New York Univ., NYO-1480-116, 1969.**[5]**David Gilbarg,*Jets and cavities*, Handbuch der Physik, Vol. 9, Part 3, Springer-Verlag, Berlin, 1960, pp. 311–445. MR**0119655****[6]**Zeev Nehari,*Conformal mapping*, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1952. MR**0045823****[7]**David Young,*Iterative methods for solving partial difference equations of elliptic type*, Trans. Amer. Math. Soc.**76**(1954), 92–111. MR**0059635**, https://doi.org/10.1090/S0002-9947-1954-0059635-7**[8]**John H. Giese,*On the truncation error in a numerical solution of the Neumann problem for a rectangle*, J. Math. and Phys.**37**(1958), 169–177. MR**0096361**, https://doi.org/10.1002/sapm1958371169**[9]**A. D. SNIDER,*Numerical Solution of Nonlinear Boundary Value Problems Using Reflection*, AEC Computing and Applied Mathematics Center, Courant Inst. Math. Sci., New York Univ., NYO-1480-167, 1971.**[10]**P. R. Garabedian,*Calculation of axially symmetric cavities and jets*, Pacific J. Math.**6**(1956), 611–684. MR**0087396**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1976-0443380-8

Keywords:
Reflection,
free boundary,
finite differences,
conformal mapping

Article copyright:
© Copyright 1976
American Mathematical Society