A Pseudospectral Fictitious Point Method for High Order Initial-Boundary Value Problems

When pseudospectral approximations are used for space derivatives, one often encounters spurious eigenvalues. These can lead to severe time stepping difficulties for PDEs. This is especially the case for equations with high order derivatives in space, requiring multiple conditions at one or both boundaries. We note here that a very simple-to-implement fictitious point approach circumvents most of these difficulties. The new approach is tested on the Kuramoto-Sivashinsky equation and on a dispersive linear PDE featuring a time-space corner singularity.

[1]  Tobin A. Driscoll,et al.  Block Pseudospectral Methods for Maxwell's Equations II: Two-Dimensional, Discontinuous-Coefficient Case , 1999, SIAM J. Sci. Comput..

[2]  Donald A. Jones,et al.  Enslaved Finite Difference Schemes for Nonlinear Dissipative PDEs , 1996 .

[3]  Bengt Fornberg,et al.  Accurate numerical resolution of transients in initial-boundary value problems for the heat equation , 2003 .

[4]  Shraiman.  Order, disorder, and phase turbulence. , 1986, Physical review letters.

[5]  Steven J. Ruuth,et al.  Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations , 1997 .

[6]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[7]  B. V. Leer,et al.  A quasi-steady state solver for the stiff ordinary differential equations of reaction kinetics , 2000 .

[8]  Chuei-Tin Chang,et al.  New insights in solving distributed system equations by the quadrature method—I. Analysis , 1989 .

[9]  Bengt Fornberg,et al.  On the Nature of Initial-Boundary Value Solutions for Dispersive Equations , 2004, SIAM J. Appl. Math..

[10]  Wilhelm Heinrichs,et al.  Spectral Approximation of Third-Order Problems , 1999, J. Sci. Comput..

[11]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[12]  Allen Taflove,et al.  Computational Electrodynamics the Finite-Difference Time-Domain Method , 1995 .

[13]  Tobin A. Driscoll,et al.  A Block Pseudospectral Method for Maxwell's Equations , 1998 .

[14]  Natasha Flyer,et al.  The Convergence of Spectral and Finite Difference Methods for Initial-Boundary Value Problems , 2001, SIAM J. Sci. Comput..

[15]  K. L. Nielsen.  Methods in numerical analysis , 1956 .

[16]  J. Hyman,et al.  THE KURAMOTO-SIV ASIDNSKY EQUATION: A BRIDGE BETWEEN POE'S AND DYNAMICAL SYSTEMS , 1986 .

[17]  Steven J. Ruuth,et al.  Implicit-explicit methods for time-dependent partial differential equations , 1995 .

[18]  Weizhang Huang,et al.  The pseudospectral method for third-order differential equations , 1992 .

[19]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[20]  Lloyd N. Trefethen,et al.  Fourth-Order Time-Stepping for Stiff PDEs , 2005, SIAM J. Sci. Comput..

[21]  Wesley H. Huang,et al.  The pseudospectral method for solving di8erential eigenvalue problems , 1994 .

[22]  R. W. Douglass,et al.  A modified tau spectral method that eliminated spurious eigenvalues , 1989 .

[23]  J. M. Keiser,et al.  A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs , 1998 .

[24]  B. Fornberg.  Generation of finite difference formulas on arbitrarily spaced grids , 1988 .

[25]  B. Fornberg.  CALCULATION OF WEIGHTS IN FINITE DIFFERENCE FORMULAS∗ , 1998 .

[26]  Bengt Fornberg,et al.  Classroom Note: Calculation of Weights in Finite Difference Formulas , 1998, SIAM Rev..

[27]  A. Siamj.,et al.  ON THE NATURE OF INITIAL-BOUNDARY VALUE SOLUTIONS FOR DISPERSIVE EQUATIONS∗ , 2003 .

[28]  Satish C. Reddy,et al.  A MATLAB differentiation matrix suite , 2000, TOMS.

[29]  Bruno Welfert.  Generation of Pseudospectral Differentiation Matrices I , 1997 .

[30]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[31]  Ronald F. Boisvert,et al.  Elimination of spurious eigenvalues in the Chebyshev Tau spectral method , 1989 .

[32]  W. Merryfield,et al.  Properties of Collocation Third-Derivative Operators , 1993 .

[33]  Wilhelm Heinrichs,et al.  Some results about the pseudospectral approximation of one-dimensional fourth-order problems , 1990 .

[34]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[35]  John P. Boyd,et al.  Compatibility conditions for time-dependent partial differential equations and the rate of convergence of Chebyshev and Fourier spectral methods , 1999 .

[36]  G. Hedstrom,et al.  Numerical Solution of Partial Differential Equations , 1966 .