The Explicit-Implicit-Null method: Removing the numerical instability of PDEs

A general method to remove the numerical instability of partial differential equations is presented. Two equal terms are added to and subtracted from the right-hand side of the PDE: the first is a damping term and is treated implicitly, the second is treated explicitly. A criterion for absolute stability is found and the scheme is shown to be convergent. The method is applied with success to the mean curvature flow equation, the Kuramoto-Sivashinsky equation, and to the Rayleigh-Taylor instability in a Hele-Shaw cell, including the effect of surface tension.

[1]  T. Hou,et al.  Removing the stiffness from interfacial flows with surface tension , 1994 .

[2]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[3]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[4]  Andrea L. Bertozzi,et al.  Axisymmetric Surface Diffusion: Dynamics and Stability of Self-Similar Pinchoff , 1998 .

[5]  W. Mullins.  Theory of Thermal Grooving , 1957 .

[6]  Colin B. Macdonald,et al.  The Implicit Closest Point Method for the Numerical Solution of Partial Differential Equations on Surfaces , 2009, SIAM J. Sci. Comput..

[7]  J. Eggers,et al.  The role of self-similarity in singularities of partial differential equations , 2008 .

[8]  Todd F. Dupont,et al.  Convergence of a step-doubling Galerkin method for parabolic problems , 1999, Mathematics of Computation.

[9]  S. Orszag.  Spectral methods for problems in complex geometries , 1980 .

[10]  Wei Lu,et al.  A Local Semi-Implicit Level-Set Method for Interface Motion , 2008, J. Sci. Comput..

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

[12]  A. Majda,et al.  Vorticity and incompressible flow , 2001 .

[13]  Jack Dongarra,et al.  Special Issue on Program Generation, Optimization, and Platform Adaptation , 2005, Proc. IEEE.

[14]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[15]  J. H. PEARCE,et al.  Functions of a Complex Variable , 1947, Nature.

[16]  Karl B Glasner,et al.  A diffuse interface approach to Hele-Shaw flow , 2003 .

[17]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

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

[19]  J. E. Glynn,et al.  Numerical Recipes: The Art of Scientific Computing , 1989 .

[20]  W. S. Edwards,et al.  Krylov methods for the incompressible Navier-Stokes equations , 1994 .

[21]  J. Conway,et al.  Functions of a Complex Variable , 1964 .

[22]  Ioannis G. Kevrekidis,et al.  Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum , 2002, SIAM J. Sci. Comput..

[23]  John R. Lister,et al.  Coalescence of liquid drops , 1999, Journal of Fluid Mechanics.

[24]  S. Kak.  The discrete Hilbert transform , 1970 .

[25]  Bernhard Kawohl,et al.  On rotationally symmetric mean curvature flow , 1991 .

[26]  Michael Shelley,et al.  A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method , 1992, Journal of Fluid Mechanics.

[27]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[28]  J. Eggers,et al.  Dripping of a crystal. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Peter Smereka,et al.  Semi-Implicit Level Set Methods for Curvature and Surface Diffusion Motion , 2003, J. Sci. Comput..

[30]  Jim Douglas,et al.  ALTERNATING-DIRECTION GALERKIN METHODS ON RECTANGLES , 1971 .