Abstract
This article establishes an error approximation of a semidiscretized scheme in the solution of a linear partial differential equation initial-valued problem. The error approximation relates the error to both equations and conditions in a problem and serves as a problem-oriented error bound. Two examples of a continuous and a discontinuous wave propagating with a speed varying in space have been studied. The results show that the propagating wave suffers distortion, and the numerical wave is not in synchronization with the exact one. This contributes to that a pointwise error measurement misleads. The discrete Fourier transform can be defined on a finite set of unequally spaced mesh points. Errors of various sources interact and cancel each other; therefore, a solution accuracy improvement can be achieved by utilizing error interaction.
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References
Ames, W. F. (1979).Numerical Methods for Partial Differential Equations, Academic Press, New York.
Bamberger, A., Euggnist, B., Helpern, L., and Joly, P. (1984). The paraxial approximation for the wave equation: Some new results,Adv. Comput. Methods Partial Differential Equations V, 340–344.
Brenner, P., and Thomas, V. (1971). Estimates near discontinuities for some difference schemes,Math. Sci. 29, 329–340.
Cambell, L. L. (1964). A comparison of the sampling theorem of Kramer and Whittaker,J. SIAM. Appl. Math. 12(1), 117–130.
Cambell, L. L. (1968). Sampling theorem for the Fourier transform of a distribution with bounded support,SIAM. J. Appl. Math. 16(3), 626–636.
Chiang, Y. F. (1981). Numerical experiments for solving symmetrix hyperbolic systems with discontinuous initial data, Ph.D. thesis, Rutgers University.
Chiang, Y. F. (1988). Truncation and accumulated errors in wave propagation problems,J. Comput. Phys. Nov., 353–372.
Chiang, Y. F. (1990). An error approximation, IMACS Proceedings of the Symposium on Approximation, Optimization, and Computing, pp. 53–58.
Chiang, Y. F. (1989). An optimal scheme for wave propagation problems, IMACS Proceedings of the International Conference on System Simulation and Scientific Computing, pp. 1–5.
Chin, Raymond C. Y. (1975). Dispersion and Gibbs phenomenon associated with difference approximation of initial-boundary value problems for hyperbolic equations,J. Comput. Phys. 18, 233–247.
Cooley, J. W., and Tukey, J. W. (1965). An algorithm for the machine calculation of complex Fourier series,Math. Comput. 19, 297–301.
Fornberg, B. (1975). On a Fourier method for the integration of hyperbolic equations,SIAM J. Num. Anal. 12, 509–528.
Giles, M. B., and Thompkins, W. T., Jr. (1985). Propagation and stability of wavelike solution of finite difference equations with variable coefficients,J. Comput. Phys. 58, 349–360.
Gottlieb, D., and Orszag, S. A. (1977). Numerical analysis of spectral methods, in CBMS Regional Conference Series, SIAM, Philadelphia.
Hamming, R. W. (1962).Numerical Methods for Scientists and Engineers, McGraw-Hill, New York.
Majda, A., and Osher, S. (1977). Propagation of errors into regions of smoothness for accurate difference approximations to hyperbolic equations,Comm. Pure Appl. Math. 671–705.
Orszag, S. A., and Jayne, L. W. (1974). Local error of difference approximations to hyperbolic equations,J. Comput. Phys. 14, 93–103.
Papoulis, H. (1962).The Fourier Integral and Its Application, McGraw-Hill, New York.
Price, H. S., and Varga, R. S. (1968). Error bounds for semidiscrete Galerkin approximations of parabolic problems with applications to petroleum reservoir mechanics, Numerical Solution of Field Problems in Continuous Physics, Proc. SIAM-AMS, Vol. 2, American Mathematical Society, Providence, 74–94.
Richter, G. R., and Falk, R. S. (1984). An analysis of a finite element method for hyperbolic equations,Adv. Comput. Methods Partial Differential Equations V, 299–301.
Richtmeyer, R. D., and Morton, K. W. (1969).Difference Methods for Initial Value Problems, Interscience Publishers, New York.
Sköllermo, G. (1975). Error analysis of finite difference schemes applied to hyperbolic initialvalued problems,Math. Comput. 33(145), 11–35.
Swarztrauber, P. N. (1984). On the spectral analysis of vector differential equations on the sphere,Adv. Comput. Methods Partial Differential Equations V, 497–501.
Trefethen, L. N. (1982). Group velocity in finite difference schemes,SIAM Rev. 24(2), 113–136.
Vichnevetsky, R. (1987). Wave propagation analysis of difference schemes for hyperbolic equations: A review,Int. J. Numer. Methods Fluids 7(5), 409–452.
Vichnevetsky, R., and Bowles, J. B. (1982).Fourier Analysis of Numerical Approximations of Hyperbolic Equations, SIAM, Philadelphia.
Vichnevetsky, R. (1981).Computer Methods for Partial Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey.
von Neumann, J., and Richtmeyer, R. D. (1950). A method for the numerical calculation of hyperdynamic shocks,J. Appl. Phys. 21, 232–237.
Whittaker, J. M. (1982). The Fourier theory of the cardinal function,Proc. R. Soc. Edinburgh 169–176.
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Chiang, YL.F. Error approximation in the solution of a linear PDE initial-valued problem. J Sci Comput 6, 283–303 (1991). https://doi.org/10.1007/BF01062814
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DOI: https://doi.org/10.1007/BF01062814