High-order accurate difference potentials methods for parabolic problems

Highly-accurate numerical methods that can efficiently handle problems with interfaces and/or problems in domains with complex geometry are crucial for the resolution of different temporal and spatial scales in many problems from physics and biology. In this paper we continue the work started in 8], and we use modest one-dimensional parabolic problems as the initial step towards the development of high-order accurate methods based on the Difference Potentials approach. The designed methods are well-suited for variable coefficient parabolic models in heterogeneous media and/or models with non-matching interfaces and with non-matching grids. Numerical experiments are provided to illustrate high-order accuracy and efficiency of the developed schemes. While the method and analysis are simpler in the one-dimensional settings, they illustrate and test several important ideas and capabilities of the developed approach.

[1]  Shan Zhao,et al.  A Matched Alternating Direction Implicit (ADI) Method for Solving the Heat Equation with Interfaces , 2014, Journal of Scientific Computing.

[2]  Sergei Utyuzhnikov,et al.  An algorithm of the method of difference potentials for domains with cuts , 2015 .

[3]  S. Tsynkov.  Numerical solution of problems on unbounded domains. a review , 1998 .

[4]  Viktor S. Ryaben’kii.  Difference potentials analogous to Cauchy integrals , 2012 .

[5]  Yekaterina Epshteyn,et al.  Algorithms composition approach based on difference potentials method for parabolic problems , 2014 .

[6]  Gunilla Kreiss,et al.  A uniformly well-conditioned, unfitted Nitsche method for interface problems , 2013 .

[7]  V S Ryaben'kii,et al.  The Numerical Example of Algorithm Composition for Solution the Boundary-Value Problems on Compound Domain Based on Difference Potentials Method. , 2003 .

[8]  Guo-Wei Wei,et al.  Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces , 2007, J. Comput. Phys..

[9]  Sergei Utyuzhnikov,et al.  Active sound control in composite regions , 2015 .

[10]  R. Fedkiw,et al.  A Boundary Condition Capturing Method for Poisson's Equation on Irregular Domains , 2000 .

[11]  Shan Zhao,et al.  WEAK GALERKIN METHODS FOR SECOND ORDER ELLIPTIC INTERFACE PROBLEMS. , 2013, Journal of computational physics.

[12]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[13]  Semyon Tsynkov,et al.  Active control of sound with variable degree of cancellation , 2009, Appl. Math. Lett..

[14]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[15]  N. K. Kulman,et al.  Method of difference potentials and its applications , 2001 .

[16]  V. Ryaben'kii.  On the Method of Difference Potentials , 2006, J. Sci. Comput..

[17]  Shan Zhao,et al.  High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources , 2006, J. Comput. Phys..

[18]  Yekaterina Epshteyn,et al.  High-order difference potentials methods for 1D elliptic type models , 2015 .

[19]  Semyon Tsynkov,et al.  Active Shielding and Control of Noise , 2001, SIAM J. Appl. Math..

[20]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[21]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[22]  Ivo Babuska,et al.  The finite element method for elliptic equations with discontinuous coefficients , 1970, Computing.

[23]  Semyon Tsynkov,et al.  Discrete Calderon's projections on parallelepipeds and their application to computing exterior magnetic fields for FRC plasmas , 2013, J. Comput. Phys..

[24]  A. Mayo.  The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions , 1984 .

[25]  Lubin G. Vulkov,et al.  The immersed interface method for two-dimensional heat-diffusion equations with singular own sources , 2007 .

[26]  Semyon Tsynkov,et al.  The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes , 2012, Journal of Scientific Computing.

[27]  RAJEN KUMAR SINHA,et al.  Optimal Error Estimates for Linear Parabolic Problems with Discontinuous Coefficients , 2005, SIAM J. Numer. Anal..

[28]  Jiangguo Liu,et al.  A matched interface and boundary method for solving multi-flow Navier-Stokes equations with applications to geodynamics , 2012, J. Comput. Phys..

[29]  Yekaterina Epshteyn,et al.  Upwind-Difference Potentials Method for Patlak-Keller-Segel Chemotaxis Model , 2012, J. Sci. Comput..

[30]  V. I. Turchaninov,et al.  Algorithm composition scheme for problems in composite domains based on the difference potential method , 2006 .

[31]  Zhilin Li,et al.  The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (Frontiers in Applied Mathematics) , 2006 .

[32]  Xu-Dong Liu,et al.  Convergence of the ghost fluid method for elliptic equations with interfaces , 2003, Math. Comput..

[33]  R. Fedkiw,et al.  A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem , 2005 .

[34]  Ronald Fedkiw,et al.  The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains , 2007, Math. Comput..

[35]  Zhilin Li,et al.  The Immersed Interface/Multigrid Methods for Interface Problems , 2002, SIAM J. Sci. Comput..

[36]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[37]  J. Zou,et al.  Finite element methods and their convergence for elliptic and parabolic interface problems , 1998 .

[38]  Randall J. LeVeque,et al.  Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension , 1997, SIAM J. Sci. Comput..

[39]  Semyon Tsynkov,et al.  Artificial boundary conditions for the numerical solution of external viscous flow problems , 1995 .

[40]  Semyon Tsynkov,et al.  A High-Order Numerical Method for the Helmholtz Equation with Nonstandard Boundary Conditions , 2013, SIAM J. Sci. Comput..