A fast Cauchy-Riemann solver

Authors:
Michael Ghil and Ramesh Balgovind

Journal:
Math. Comp. **33** (1979), 585-635

MSC:
Primary 65F05; Secondary 65N15

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521278-7

MathSciNet review:
521278

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Abstract | References | Similar Articles | Additional Information

Abstract: We present a solution algorithm for a second-order accurate discrete form of the inhomogeneous Cauchy-Riemann equations. The algorithm is comparable in speed and storage requirements with fast Poisson solvers. Error estimates for the discrete approximation of sufficiently smooth solutions of the problem are established; numerical results indicate that second-order accuracy obtains even for solutions which do not have the required smoothness. Different combinations of boundary conditions are considered and suitable modifications of the solution algorithm are described and implemented.

**[1]**Eduard Batschelet,*Über die numerische Auflösung von Ranswertproblemen bei elliptischen partiellen Differentialgleichungen*, Z. Angew. Math. Physik**3**(1952), 165–193 (German). MR**0060912****[2]**J. H. Bramble and B. E. Hubbard,*Approximation of derivatives by finite difference methods in elliptic boundary value problems*, Contributions to Differential Equations**3**(1964), 399–410. MR**0166935****[3]**J. H. Bramble and B. E. Hubbard,*A finite difference analogue of the Neumann problem for Poisson’s equation*, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.**2**(1965), 1–14. MR**0191107****[4]**O. BUNEMAN,*A Compact Non-Iterative Poisson Solver*, SUIPR Report No. 294, Inst. Plasma Research, Stanford Univ., May 1969, 11 pp.**[5]**B. L. Buzbee, G. H. Golub, and C. W. Nielson,*On direct methods for solving Poisson’s equations*, SIAM J. Numer. Anal.**7**(1970), 627–656. MR**0287717**, https://doi.org/10.1137/0707049**[6]**Lothar Collatz,*The numerical treatment of differential equations. 3d ed*, Translated from a supplemented version of the 2d German edition by P. G. Williams. Die Grundlehren der mathematischen Wissenschaften, Bd. 60, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR**0109436****[7]**J. W. COOLEY, P. A. W. LEWIS & P. D. WELCH, "The finite Fourier transform,"*IEEE Trans. Audio and Electroacoustics*, v. 17, 1969, pp. 77-85.**[8]**E. G. D′jakonov,*On certain iterative methods for solving nonlinear difference equations*, Conference on Numerical Solution of Differential Equations (Dundee, 1969), Springer, Berlin, 1969, pp. 7–22. MR**0323134****[9]**Fred W. Dorr,*The direct solution of the discrete Poisson equation on a rectangle.*, SIAM Rev.**12**(1970), 248–263. MR**0266447**, https://doi.org/10.1137/1012045**[10]**T. ELVIUS & A. SUNDSTRÖM, "Computationally efficient schemes and boundary conditions for a fine-mesh barotropic model based on the shallow-water equations,"*Tellus*, v. 25, 1973, pp. 132-156.**[11]**George E. Forsythe and Wolfgang R. Wasow,*Finite-difference methods for partial differential equations*, Applied Mathematics Series, John Wiley & Sons, Inc., New York-London, 1960. MR**0130124****[12]**D. Fischer, G. Golub, O. Hald, C. Leiva, and O. Widlund,*On Fourier-Toeplitz methods for separable elliptic problems*, Math. Comp.**28**(1974), 349–368. MR**0415995**, https://doi.org/10.1090/S0025-5718-1974-0415995-2**[13]**S. GERSCHGORIN; "Fehlerabschätzung für das Differenzverfahren zur Lösung partieller Differentialgleichungen,"*Z. Angew. Math. Mech.*, v. 10, 1930, pp. 373-382.**[14]**M. GHIL, "The initialization problem in numerical weather prediction," in*Improperly Posed Boundary Value Problems*(A. Carasso and A. P. Stone, Eds.), Research Notes in Math., vol. 1, Pitman, London, 1975, pp. 105-123.**[15]**M. GHIL,*Initialization by Compatible Balancing*, Report 75-16, Inst. Comp. Appl. Sci. Engr., Hampton, Virginia, 1975, 38 pp.**[16]**M. GHIL & B. SHKOLLER, "Wind laws for shockless initialization,"*Ann. Meteor.*(Neue Folge), v. 11, 1976, pp. 112-115.**[17]**M. GHIL, B. SHKOLLER & V. YANGARBER, "A balanced diagnostic system compatible with a barotropic prognostic model,"*Mon. Wea. Rev.*, v. 105, 1977, pp. 1223-1238.**[18]**G. GOLUB, "Direct methods for solving elliptic difference equations," in*Symposium on the Theory of Numerical Analysis*(J. Ll. Morris, Ed.), Lecture Notes in Math., vol. 193, Springer-Verlag, Berlin, 1971, pp. 1-19.**[19]**James E. Gunn,*The solution of elliptic difference equations by semi-explicit iterative techniques*, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.**2**(1965), 24–45. MR**0179962****[20]**Bertil Gustafsson,*An alternating direction implicit method for solving the shallow water equations*, J. Computational Phys.**7**(1971), 239–254. MR**0282548****[21]**R. W. Hockney,*A fast direct solution of Poisson’s equation using Fourier analysis*, J. Assoc. Comput. Mach.**12**(1965), 95–113. MR**0213048**, https://doi.org/10.1145/321250.321259**[22]**R. W. HOCKNEY, "The potential calculation and some applications," in*Methods in Computational Physics*(B. Adler, S. Fernbach and M. Rotenberg, Eds.), vol. 9 (Plasma Physics), Academic Press, New York, 1969, pp. 135-211.**[23]**W. E. LANGLOIS,*Vorticity-Stream Function Computation of Incompressible Fluid Flow with an Almost-Flat Free Surface*, IBM Research Report RJ 1794 (#26092), 1976, 8 pp.**[24]**H. LOMAX & E. D. MARTIN, "Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations,"*J. Computational Phys.*, v. 15, 1974, pp. 55-80.**[25]**E. D. MARTIN & H. LOMAX,*Rapid Finite-Difference Computation of Subsonic and Transonic Aerodynamic Flows*, AIAA Paper No. 74-11, 1974, 13 pp.**[26]**E. D. MARTIN & H. LOMAX,*Variants and Extensions of a Fast Direct Numerical Cauchy-Riemann Solver, with Illustrative Applications*, NASA Tech. Note TN D-7934, 1977, 94 pp.**[27]**Joseph Oliger and Arne Sundström,*Theoretical and practical aspects of some initial boundary value problems in fluid dynamics*, SIAM J. Appl. Math.**35**(1978), no. 3, 419–446. MR**0521943**, https://doi.org/10.1137/0135035**[28]**Patrick J. Roache,*Computational fluid dynamics*, Hermosa Publishers, Albuquerque, N.M., 1976. With an appendix (“On artificial viscosity”) reprinted from J. Computational Phys. 10 (1972), no. 2, 169–184; Revised printing. MR**0411358****[29]**U. Schumann and Roland A. Sweet,*A direct method for the solution of Poisson’s equation with Neumann boundary conditions on a staggered grid of arbitrary size*, J. Computational Phys.**20**(1976), no. 2, 171–182. MR**0395258****[30]**Paul N. Swarztrauber,*A direct method for the discrete solution of separable elliptic equations*, SIAM J. Numer. Anal.**11**(1974), 1136–1150. MR**0368399**, https://doi.org/10.1137/0711086**[31]**Roland A. Sweet,*A generalized cyclic reduction algorithm*, SIAM J. Numer. Anal.**11**(1974), 506–520. MR**0520169**, https://doi.org/10.1137/0711042**[32]**O. WIDLUND, "On the use of fast methods for separable finite-difference equations for the solution of general elliptic problems," in*Sparse Matrices and Their Applications*(D. J. Rose and R. A. Willoughby, Eds.), Plenum Press, New York, 1972, pp. 121-131.**[33]**Olof Widlund,*Capacitance matrix methods for Helmholtz’s equation on general bounded regions*, Numerical treatment of differential equations (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1976) Springer, Berlin, 1978, pp. 209–219. Lecture Notes in Math., Vol. 631. MR**0474873**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521278-7

Keywords:
Fast direct solvers,
Cauchy-Riemann equations,
elliptic first-order systems,
transonic flow

Article copyright:
© Copyright 1979
American Mathematical Society