Bibliography for the Finite Difference Method for PDE's

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  1. A fourth-order finite-difference method for acoustic wave equation on irregular grids
    Oliveira, S. A. M.
    Geophysics, 2003, vol. 68, no. 2, pp. 672-676, Ingenta.
  2. The Homogeneous Finite-Difference Formulation of the P-SV-Wave Equation of Motion
    Slawinski, R. S.; Krebes, E. S.
    Studia Geophysica et Geodaetica, 2002, vol. 46, no. 4, pp. 731-751, Ingenta.
  3. Adaptive high-order finite-difference method for nonlinear wave problems.    
    Fatkullin, I.; Hesthaven, J. S.    
    J. Sci. Comput. 16 (2001), no. 1, 47--67, MathSciNet.  
  4. ADI plus interpolation: Accurate finite-difference solution to 3D paraxial wave equation
    Wang, Y.
    Geophysical Prospecting, v 49, n 5, September, 2001, p 547-556, Compendex.
  5. Comparison of high-accuracy finite-difference methods for linear wave propagation.    
    Zingg, David W.    
    SIAM J. Sci. Comput. 22 (2000), no. 2, 476--502 (electronic), MathSciNet.  
  6. Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square.
    Zuazua, E.
    J. Math. Pures Appl. (9) 78 (1999), no. 5, 523--563, MathSciNet.  
  7. Finite difference scheme for traveling wave solutions to Burgers equation
    Mickens, Ronald E.
    Numerical Methods for Partial Differential Equations, v 14, n 6, Nov, 1998, p 815-820, Compendex.
  8. Boundary conditions for the finite difference beam propagation method based on plane wave solutions of the Fresnel equation
    Lohmeyer, Manfred; Shamonin, Mikhail; Hertel, Peter
    IEEE Journal of Quantum Electronics, v 33, n 2, Feb, 1997, p 279-285, Compendex.
  9. Construction and analysis of Fourth-Order Finite Difference Schemes for the Acoustic Wave Equation in Nonhomogeneous Media  
    Gary Cohen, Patrick Joly
    SIAM Journal on Numerical Analysis, Vol. 33, No. 4. (Aug., 1996), pp. 1266-1302, Jstor.  
  10. A Block Finite Difference Scheme for Second-Order Elliptic Problems with Discontinuous Coefficients  
    Jian Shen  
    SIAM Journal on Numerical Analysis, Vol. 33, No. 2. (Apr., 1996), pp. 686-706, Jstor.  
  11. Difference approximations for wave equations via finite elements. I. Construction and performance
    Card, Curtis L.; Allen, Myron B.
    Numerical Methods for Partial Differential Equations, v 11, n 2, Mar, 1995, p 127, Compendex.
  12. Exact finite difference scheme for a spherical wave equation
    Mickens, R.E.
    Journal of Sound and Vibration, v 182, n 2, Apr 27, 1995, p 342, Compendex.
  13. Effective filtering of artifacts for implicit finite-difference paraxial wave equation migration
    Bunks, C.
    Geophysical Prospecting, v 43, n 2, Feb, 1995, p 203-220, Compendex.
  14. Finite-difference method for the third-order simplified wave equation: assessment and application
    Lu, Zhang-Ning; Bansal, Rajeev
    IEEE Transactions on Microwave Theory and Techniques, v 42, n 1, Jan, 1994, p 132-136, Compendex.
  15. Outgoing Boundary Conditions for Finite-Difference Elliptic Water-Wave Models  
    Bingyi Xu; Vijay Panchang  
    Proceedings: Mathematical and Physical Sciences, Vol. 441, No. 1913. (Jun. 8, 1993), pp. 575-588
  16. An Error Estimate for a Finite Difference Scheme Approximating a Hyperbolic System of Conservation Laws  
    Aslak Tveito, Ragnar Winther
    SIAM Journal on Numerical Analysis, Vol. 30, No. 2. (Apr., 1993), pp. 401-424, Jstor.  
  17. Accurate finite-difference operators for modelling the elastic wave equation
    Jastram, Cord; Behle, Alfred
    Geophysical Prospecting, v 41, n 4, May, 1993, p 453-458, Compendex.
  18. An eigenvalue problem for a system of finite difference equations approximating a linear water wave equation.
    Matsuki, Mihoko; Ushijima, Teruo
    Japan J. Indust. Appl. Math. 9 (1992), no. 1, 91--116, MathSciNet.
  19. Jacobi Iteration in Implicit Difference Schemes for the Wave Equation   
    D. B. Duncan, M. A. M. Lynch   
    SIAM Journal on Numerical Analysis, Vol. 28, No. 6. (Dec., 1991), pp. 1661-1679, Jstor.  
  20. Convergence of Finite Difference Schemes for Conservation Laws in Several Space Dimensions: The Corrected Antidiffusive Flux Approach  
    Frederic Coquel; Philippe Le Floch  
    Mathematics of Computation, Vol. 57, No. 195. (Jul., 1991), pp. 169-210, Jstor.  
  21. Wave propagation analysis for finite difference solutions of the three-dimensional tidal equations
    Stevens, M.W.; Noye, B.J. Source:
    Computers & Fluids, v 19, n 1, 1991, p 75-91, Compendex.
  22. A generalized image principle for the wave equation with absorbing boundary conditions and applications to fourth order finite difference schemes.
    Ha-Duong, Tuong; Joly, Patrick
    Third International Conference on Hyperbolic Problems, Vol. I, II (Uppsala, 1990), 528--541, Studentlitteratur, Lund, 1991, MathSciNet.
  23. Finite Difference Method for a Two-Sex Model of Population Dynamics  
    Todd Arbogast; Fabio A. Milner  
    SIAM Journal on Numerical Analysis, Vol. 26, No. 6. (Dec., 1989), pp. 1474-1486, Jstor.  
  24. Nonlinear Instability in Dissipative Finite Difference Schemes  
    Andrew Stuart  
    SIAM Review, Vol. 31, No. 2. (Jun., 1989), pp. 191-220, Jstor.  
  25. An Analysis of a Uniformly Convergent Finite Difference/Finite Element Scheme for a Model Singular-Perturbation Problem  
    Eugene C. Gartland, Jr.  
    Mathematics of Computation, Vol. 51, No. 183. (Jul., 1988), pp. 93-106, Jstor.  
  26. Stability of Finite Difference Schemes for Two-Point Boundary Value Problems  
    C. De Boor; F. De Hoog  
    SIAM Journal on Numerical Analysis, Vol. 23, No. 5. (Oct., 1986), pp. 925-935, Jstor.  
  27. Stability of Finite-Difference Models Containing Two Boundaries or Interfaces
    Lloyd N. Trefethen
    Mathematics of Computation, Vol. 45, No. 172. (Oct., 1985), pp. 279-300, Jstor.  
  28. Exact finite difference schemes for the nonlinear unidirectional wave equation.
    Mickens, R. E.
    J. Sound Vibration 100 (1985), no. 3, 452--455, MathSciNet.
  29. Error Estimates for Finite Difference Approximations to Hyperbolic Equations for Large Time  
    William Layton  
    Proceedings of the American Mathematical Society, Vol. 92, No. 3. (Nov., 1984), pp. 425-431, Jstor.  
  30. Two New Finite Difference Schemes for Parabolic Equations  
    J. R. Cash  
    SIAM Journal on Numerical Analysis, Vol. 21, No. 3. (Jun., 1984), pp. 433-446, Jstor.  
  31. Stability Analysis of Finite Difference Schemes for the Advection-Diffusion Equation  
    Tony F. Chan  
    SIAM Journal on Numerical Analysis, Vol. 21, No. 2. (Apr., 1984), pp. 272-284., Jstor.  
  32. On a Class of High Resolution Total-Variation-Stable Finite-Difference Schemes  
    Ami Harten, Peter D. Lax
    SIAM Journal on Numerical Analysis, Vol. 21, No. 1. (Feb., 1984), pp. 1-23, Jstor.  
  33. Applications of energy methods to finite-difference solutions of the parabolic wave equation.
    McDaniel, Suzanne T.
    Computational ocean acoustics (New Haven, Conn., 1984).  Comput. Math. Appl.  11  (1985),  no. 7-8, 823--829, MathSciNet.
  34. On Convergence of Monotone Finite Difference Schemes with Variable Spatial Differencing  
    Richard Sanders  
    Mathematics of Computation, Vol. 40, No. 161. (Jan., 1983), pp. 91-106, Jstor.  
  35. On Numerical Boundary Treatment of Hyperbolic Systems for Finite Difference and Finite Element Methods  
    David Gottlieb; Max Gunzburger; Eli Turkel  
    SIAM Journal on Numerical Analysis, Vol. 19, No. 4. (Aug., 1982), pp. 671-682, Jstor.  
  36. Group Velocity in Finite Difference Schemes  
    Lloyd N. Trefethen  
    SIAM Review, Vol. 24, No. 2. (Apr., 1982), pp. 113-136, Jstor.  
  37. Compact Finite Difference Schemes for Mixed Initial-Boundary Value Problems  
    Richard B. Philips, Milton E. Rose
    SIAM Journal on Numerical Analysis, Vol. 19, No. 4. (Aug., 1982), pp. 698-720, Jstor.  
  38. A finite-difference treatment of interface conditions for the parabolic wave equation: the horizontal interface.
    McDaniel, Suzanne T.; Lee, Ding
    J. Acoust. Soc. Amer. 71 (1982), no. 4, 855--858, MathSciNet.
  39. On a Fourth Order Accurate Implicit Finite Difference Scheme for Hyperbolic Conservation Laws: I. Nonstiff Strongly Dynamic Problems  
    Amiram Harten; Hillel Tal-Ezer  
    Mathematics of Computation, Vol. 36, No. 154. (Apr., 1981), pp. 353-373, Jstor.  
  40. Uniform Expansions for a Class of Finite Difference Schemes for Elliptic Boundary Value Problems  
    Harry Munz  
    Mathematics of Computation, Vol. 36, No. 153. (Jan., 1981), pp. 155-170, Jstor.  
  41. Finite-difference solution to the parabolic wave equation.
    Lee, Ding; Botseas, George; Papadakis, John S.
    J. Acoust. Soc. Amer. 70 (1981), no. 3, 795--800, MathSciNet.
  42. Computation of Steady Shocks by Second-Order Finite-Difference Schemes  
    Lasse K. Karlsen  
    Mathematics of Computation, Vol. 34, No. 150. (Apr., 1980), pp. 391-400, Jstor.  
  43. Stability Restrictions on Second Order, Three Level Finite Difference Schemes for Parabolic Equations  
    J. M. Varah  
    SIAM Journal on Numerical Analysis, Vol. 17, No. 2. (Apr., 1980), pp. 300-309, Jstor.  
  44. Error Analysis of Finite Difference Schemes Applied to Hyperbolic Initial Boundary Value Problems  
    Gunilla Skollermo  
    Mathematics of Computation, Vol. 33, No. 145. (Jan., 1979), pp. 11-35, Jstor.  
  45. A Finite Difference Scheme for a System of Two Conservation Laws with Artificial Viscosity  
    David Hoff  
    Mathematics of Computation, Vol. 33, No. 148. (Oct., 1979), pp. 1171-1193, Jstor.  
  46. A finite difference scheme for the regularized long wave equation.
    Goda, Katuhiko
    Numerical analysis of evolution equations (Kyoto, 1978), pp. 135--146, Lecture Notes Numer. Appl. Anal., 1, Kinokuniya Book Store, Tokyo, 1979, MathSciNet.
  47. An implicit, compact, finite difference method to solve hyperbolic equations.    
    Wirz, H. J.; De Schutter, F.; Turi, A.    
    Math. Comput. Simulation 19 (1977), no. 4, 241--261, MathSciNet.   
  48. The Relative Efficiency of Finite Difference and Finite Element Methods. I: Hyperbolic Problems and Splines  
    Blair Swartz; Burton Wendroff  
    SIAM Journal on Numerical Analysis, Vol. 11, No. 5. (Oct., 1974), pp. 979-993, Jstor.  
  49. A fully implicit finite difference approximation to the one-dimensional wave equation using a cubic spline technique.
    Raggett, G. F.; Wilson, P. D.
    J. Inst. Math. Appl. 14 (1974), 75--77, MathSciNet.
  50. Higher Order Accuracy Finite Difference Algorithms for Quasi-Linear, Conservation Law Hyperbolic Systems  
    S. Abarbanel; D. Gottlieb  
    Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 505-523, Jstor.  
  51. A Note on the Stability of an Iterative Finite-Difference Method for Hyperbolic Systems   
    Moshe Goldberg  
    Mathematics of Computation, Vol. 27, No. 121. (Jan., 1973), pp. 41-44, Jstor.  
  52. Stability of a Finite Difference Scheme with "Wrong" Boundary Conditions  
    D. E. Koster  
    SIAM Journal on Numerical Analysis, Vol. 10, No. 6. (Dec., 1973), pp. 1039-1046, Jstor.  
  53. A Characteristic Finite Difference Method for the Wave Equation in Two Dimensions  
    C. M. Ablow
    SIAM Journal on Numerical Analysis, Vol. 9, No. 1. (Mar., 1972), pp. 152-164, Jstor.  
  54. Convergent Finite Difference Schemes for Nonlinear Parabolic Equations  
    Albert C. Reynolds, Jr.  
    SIAM Journal on Numerical Analysis, Vol. 9, No. 4. (Dec., 1972), pp. 523-533, Jstor.  
  55. The general solution of some finite difference equations analogous to the wave equation.
    McKiernan, M. A.
    Aequationes Math. 8 (1972), 263--266, MathSciNet.
  56. A Finite Difference Scheme and an Existence Theorem for a Nonlinear Hyperbolic System of Differential Equations  
    Pierre Jamet  
    SIAM Journal on Numerical Analysis, Vol. 8, No. 3. (Sep., 1971), pp. 524-535, Jstor.  
  57. Some Recent Applications of Asymptotic Error Expansions to Finite-Difference Schemes  
    O. B. Widlund  
    Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 323, No. 1553, A Discussion on Numerical Analysis of Partial Differential Equations. (Jun. 8, 1971), pp. 167-177, Jstor.  
  58. An Iterative Finite-Difference Method for Hyperbolic Systems  
    S. Abarbanel, G. Zwas  
    Mathematics of Computation, Vol. 23, No. 107. (Jul., 1969), pp. 549-565, Jstor.  
  59. Asymptotic Behavior of Solutions to the Finite-Difference Wave Equation  
    Carl E. Pearson
    Mathematics of Computation, Vol. 23, No. 108. (Oct., 1969), pp. 711-715, Jstor.  
  60. Generalized Finite-Difference Schemes  
    Blair Swartz; Burton Wendroff  
    Mathematics of Computation, Vol. 23, No. 105. (Jan., 1969), pp. 37-49, Jstor.  
  61. Finite-Difference Methods for Nonlinear Hyperbolic Systems. II  
    A. R. Gourlay; J. Ll. Morris  
    Mathematics of Computation, Vol. 22, No. 103. (Jul., 1968), pp. 549-556, Jstor.  
  62. Finite-Difference Methods for Nonlinear Hyperbolic Systems  
    A. R. Gourlay; J. Ll. Morris  
    Mathematics of Computation, Vol. 22, No. 101. (Jan., 1968), pp. 28-39, Jstor.  
  63. Some Remarks on the Lax-Wendroff Finite-Difference Scheme for Nonsymmetric Hyperbolic Systems  
    Masaya Yamaguti  
    Mathematics of Computation, Vol. 21, No. 100. (Oct., 1967), pp. 611-619, Jstor.  
  64. On the Construction of Consistent Finite Difference Schemes with Certain Invariant Subspaces  
    Dennis Eisen  
    SIAM Journal on Numerical Analysis, Vol. 4, No. 3. (Sep., 1967), pp. 349-356, Jstor.  
  65. Stability and Convergence of Finite Difference Schemes with Singular Coefficients  
    Dennis Eisen  
    SIAM Journal on Numerical Analysis, Vol. 3, No. 4. (Dec., 1966), pp. 545-552, Jstor.  
  66. A Generalization of the Lax-Richtmyer Theorem on Finite Difference Schemes  
    John Gary  
    SIAM Journal on Numerical Analysis, Vol. 3, No. 3. (Sep., 1966), pp. 467-473, Jstor.  
  67. On Certain Finite Difference Schemes for Hyperbolic Systems  
    John Gray  
    Mathematics of Computation, Vol. 18, No. 85. (Jan., 1964), pp. 1-18, Jstor.  
  68. Finite Difference Schemes for Differential Equations  
    Milton E. Rose  
    Mathematics of Computation, Vol. 18, No. 86. (Apr., 1964), pp. 179-195, Jstor.  
  69. The Structure of Certain Finite Difference Schemes  
    Burton Wendroff  
    SIAM Review, Vol. 3, No. 3. (Jul., 1961), pp. 237-242, Jstor.  
  70. Error Bounds in Finite-Difference Approximation to Solutions of Symmetric Hyperbolic Systems  
    H. F. Weinberger  
    Journal of the Society for Industrial and Applied Mathematics, Vol. 7, No. 1. (Mar., 1959), pp. 49-75, Jstor.  
  71. On the behavior of a solution of a finite-difference analogue of the wave equation. (Russian)
    Kamynin, L. I.
    Prikl. Mat. Meh. 19 (1955), 589--598, MathSciNet.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004