A class of finite difference schemes for interface problems with an HOC approach

TL;DR: In this article, the authors proposed a new methodology for numerically solving elliptic and parabolic equations with discontinuous coefficients and singular source terms, obtained by clubbing a recently developed Higher Order Compact (HOC) methodology with special interface treatment for the points just next to the points of discontinuity.

Abstract: In this paper, we propose a new methodology for numerically solving elliptic and parabolic equations with discontinuous coefficients and singular source terms. This new scheme is obtained by clubbing a recently developed Higher Order Compact (HOC) methodology with special interface treatment for the points just next to the points of discontinuity. The overall order of accuracy of the scheme is at least second. We first formulate the scheme for one-dimensional (1D) problems and then extend it directly to two-dimensional (2D) problems in polar coordinates. In the process, we also perform convergence and related analysis for both the cases. Finally, we show a new direction of implementing the methodology to 2D problems in cartesian coordinates. We then conduct numerous numerical studies on a number of problems, both for 1D and 2D cases including the flow past circular cylinder governed by the incompressible Navier-Stokes (N-S) equations. We compare our results with existing numerical and experimental results. In all the cases our formulation is found to produce better results on relatively coarser grids. For the circular cylinder problem, the scheme used is seen to capture all the flow characteristics including the famous von-Karman vortex street. Copyright c © 2010 John Wiley & Sons, Ltd.

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
Int. J. Numer. Meth. Fluids 2010; 00:1–37
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld
A Class Of Finite Difference Schemes For Interface Prob lems
With An HOC Approach
H. V. R. Mittal
1
, Jiten C. Kalita
2
, Rajendra K. Ray
1∗
1
School of Basic Sciences, Indian Institute of Technology Mandi, India.
2
Department of Mathematics, Indian Institute of Technology Guwahati, India.
SUMMARY
In this paper, we propose a new methodology for numerically solving elliptic and parabolic equations with
discontinuous coefficients and singular source terms. This new scheme is obtained by clubbing a recently
developed Higher Order Compact (HOC) methodology with special interface treatment for the points just
next to the points of discontinuity. The overall order of accuracy of the scheme is at least second. We first
formulate the scheme for one-dimensional (1D) problems and then extend it directly to two-dimensional
(2D) problems in polar coordinates. In the process, we also perform convergence and related analysis for
both the cases. Finally, we show a new direction of implementing the methodology to 2D problems i n
cartesian coordinates. We then conduct numerous numerical studies on a number of problems, both for 1D
and 2D cases including the flow past circular cylinder governed by the incompressible Navier-Stokes (N-S)
equations. We compare our res ults with existing numerical and experimental results. In all the cases our
formulation is found to produce better res ults on relatively coarser grids. For the circular cylinder problem,
the scheme used is seen to capture all the flow characteristics including t he famous von-K´arm´an vortex
street. Copyright
c
 2010 John Wiley & Sons, Ltd.
Received . . .
KEY WORDS: Interface; HOC scheme; Navier-Stokes equation; discontinuous coefficients; non-
uniform grids; von-K´arm´an vortex street.
1. INTRODUCTION
Flow problems involving discon tinuous coefficients and singular source terms still pose a tough
challenge in the field of computational fluid dynam ics (CFD). Such problems g e nerally have
non smooth solutions with jumps across lower dimensional in terfaces. Standard finite difference
approximations fail t o yield correct numerical solutions of such problems because of the fact that
the Taylor’s expansions, which provide the platforms for such approximations, are not valid for non
smooth functions. Of l ate, there has been a spur of i nterest in developing numerical methods for
computing mult iphase flows and multi-physics problems with interfaces. Such problem s arise in
numerous branches of science and engineering such as biochemical processing, free surface flow,
drop deformation, solid mechanics, porous media flow, formation of gas bubbles in liquid, mining
etc. For example, in problems involving dissimilar materials, at the i nterfaces, the m aterial properties
(elastic moduli, permeability, conductivity, etc) are discontinuous.
The last few decades have seen several approaches for numerically solving the interface equat ions
[
18, 19, 20, 31, 39, 40, 45]. Mo st of the earlier numerical works on such problems involved mainl y
the use of immersed boundary (IBM) or immersed interface (IIM) methods on uniform grids and
∗
Correspondence to: School of Basic Sciences, Indian Institute of Technology Mandi, India
Copyright
c
 2010 John Wiley & Sons, Ltd.
Prepared using fldauth.cls [Version: 2010/05/13 v2.00]