1.
Introduction
Interface problems arise in many
applications. For example, when there are two different materials, such as a
?uid ?ow and a porous media, or the same material but at different states, such
as water and ice, we are dealing with an interface problem. If ordinary or partial
differential equations are used to model multi-phase ?ows, multi-physics
problems, the parameters in the governing differential equations are typically
discontinuous, or even the governing PDEs are different as in the Stokes and
Darcy’s coupling, and the source terms are often singular to re?ect source/sink
distributions along co-dimensional interfaces. Because of these irregularities,
the solutions to the differential equations are typically non-smooth, or even
discontinuous as in the example of the pressure inside and outside an in?ated
balloon. As a result, many standard numerical methods based on the assumption
of smoothness of solutions do not work or work poorly for interface problems.
With some initial and boundary
conditions, where is an interface, see Figure 1 for an illustration. We assume that
the coefficient ß has
a ?nite jump across the interface corresponding to two different materials or
states and with a source term along the interface as in Peskin’s Immersed
Boundary model (Peskin & McQueen,1995). It is known that the IB method is a
smoothing method and has shown to be ?rst order accurate by this author and
other scholars, see (Li, 2015; Mori, 2018).
Where the second jump is the flux jump and n is the normal direction with |n|=1
pointing to the ?^{+} side.
For a regular heat equation, the
alternating direction implicit (ADI) method using a ?nite difference (FD)
discretization is very efficient for time-dependent parabolic PDEs, see for
example (Li, Qiao, & Tang, 2017). The classical ADI method is second-order
accurate and unconditionally stable if the standard 3-point central ?nite
difference discretization is used in each coordinate direction. An ADI method
only requires to solve a sequence of tridiagonal systems of equations which can
be done in O(N^{2}) operations, and is considered as one of the most
efficient algorithms.
Figure 2: Snap contour
plots of the computed solution with N =
160 of a ?ow through a porous media.
In Figure 2, we show plots of the streamlines of the ?ow from left passes
different objects with different permeability. For particles with large
permeability, the ?ow gets saturated while the ?ow goes around if the
permeability is very small.
2.
A brief review of augmented IIM
There
are at least two motivations for using augmented strategies. The ?rst one is to
get a faster algorithm compared to a direct discretization, particularly to
take advantage of existing fast solvers. The second one (more important one) is
that, for some interface problems, an augmented approach may be the only way to
derive an accurate algorithm. The augmented techniques enable us to decouple
the jump conditions so that the idea of the immersed interface method can be
applied, see for example, (Li, 2015; Li, Ito, & Lai, 2007; Li & Qin,
2017).
The application of the augmented
approach for two dimensional interior irregular domains can be found in (Li,
Zhao, & Gao,1999) for simulating electro-migration of voids in an integrated
circuit; in (Hunter, Li, & Zhao, 2002 ) for simulating the motion of drops;
in (Li & Wang, 2003) for solving Navier-Stokes equations on irregular
domains using the stream- vorticity formulation; in (Jin & Wang, 2002) for
micro-structure evolution in the chemical vapor in?ltration process, and (Chen,
Li, & Lin, 2008) for solving bi-harmonic
equations on irregular domains; and recently work for bi-harmonic interface
problems (Li & Qin, 2017). The augmented approach for generalized Helmholtz
equations has also been developed in three dimensions and has been applied to
an inverse problem of shape optimization in (Deng, Ito, & Li, 2003; Ito,
Kunisch, & Li, 2001). While augmented methods have some similarities to
boundary integral methods or integral equation approaches to ?nd a source
strength, the augmented methods have a few special features: (1) no Green
function is needed, and therefore there is no need to evaluate singular
integrals; (2) there is no need to set up the system of equations for the
augmented variable explicitly; (3) they are applicable to general PDEs with or
without source terms; (4) the process does not depend on the boundary
conditions. On the other hand, we may have less knowledge about the condition number
of the Schur complement system and how to apply pre-conditioning techniques.
Figure 3: Simulations
with the augmented IIM. (a): An
interface with corners in a cavity ?ow, see (Li, Lai, Peng, & Zhang, 2018).
(b): An irregular domain problem: a
?ow passes through a stationary cylinder computed with Re = 200, see (Ito, Lai, & Li, 2009).
Recent
a remarkable progress on augmented IIM is the convergence proof and for
variable coefficients. It has been shown in (Li, Ji, & Chen, 2016) that not
only the computed solution is second order accurate globally in the strongest
maximum norm, but also the computed gradient from each side of the interface.
The method is designed for closed smooth interfaces not for open-ended
interface such as cracks. For closed interfaces but with corners, see the left
plot in Figure 3. The convergence of
the augmented method for elliptic interface problems has been shown both in one
and two dimensions under appropriate regularity assumptions and a piecewise
constant ß(x). For a variable coefficient ß(x), the conclusions are still true if h is small enough, that is, in the
asymptotic sense.
New
applications of the augmented IIM include multi-physics simulations with
different governing equations on different regions such as the Stokes or
Navier-Stokes equations coupled with the Darcy’s law in (Li, 2015; Li, Lai,
Peng, & Zhang, 2018), determine the crank tips for a Mumford-Shah minimizer
of a free boundary value problem in (Li & Mikayelyan, 2016); the 3D
compressible bubbles in a incompressible ?uid (Li, Xiao, Cai, Zhao, & Luo,
2015); and an efficient preconditioner for the Schur complement matrix resulted
from AIIM in (Angot & Li, 2017; Xia, Li, & Ye, 2015).
References
Angot, P., & Li, Z. L. (2017). An augmented IIM &
preconditioning technique for jump embedded boundary conditions. International Journal of Numerical Analysis
and Modeling, 14(4-5), 712-729.
Chen, G., Li,
Z. L., & Lin, P. (2008). A fast finite difference method for biharmonic
equations on irregular domains and its application to an incompressible Stokes
flow. Advances in Computational
Mathematics, 29(2), 113-133.
Deng, S. Z.,
Ito, K., & Li, Z. L. (2003). Three-dimensional elliptic solvers for
interface problems and applications. Journal
of Computational Physics, 184(1), 215-243.
Hunter, J. K.,
Li, Z. L., & Zhao, H. K. (2002). Reactive autophobic spreading of drops. Journal of Computational Physics,
183(2), 335-366.
Ito, K.,
Kunisch, K., & Li, Z. L. (2001). Level-set function approach to an inverse
interface problem. Inverse Problems,
17(5), 1225-1242.
Ito, K., Lai,
M. C., & Li, Z. (2009). A well-conditioned augmented system for solving
Navier-Stokes equations in irregular domains. Journal of Computational Physics, 228(7), 2616-2628.
Jin, S. & Wang,
X. (2002), Robust numerical simulation of porosity evolution in chemical vapor
in?ltration, Journal of Computational
Physics, 179, 557–577.
Li, Z. L.
(2016). An augmented Cartesian grid method for Stokes-Darcy fluid-structure
interactions. International Journal for
Numerical Methods in Engineering, 106(7), 556-575.
Li, Z. L.
(2015). On convergence of the immersed boundary method for elliptic interface
problems, Math. Comp. 84 (293), 1169–1188.
Z. Li, K. Ito,
and M-C. Lai, An augmented approach for Stokes equations with a discontinuous
viscosity and singular forces, Computers
and Fluids, 36 (2007), 622–635.
Li, Z. L., Ji, H. F., & Chen, X. H.
(2017). Accurate solution and gradient computation for elliptic interface
problems with variable coefficients. SIAM
Journal on Numerical Analysis, 55(2), 570-597.
Li, Z. L.,
Lai, M. C., Peng, X. F., & Zhang, Z. Y. (2018). A least squares augmented
immersed interface method for solving Navier-Stokes and Darcy coupling
equations. Computers & Fluids,
167, 384-399.
Li, Z. L. &
A. Mayo (1993), ADI methods for heat equations with discontinuities along an
arbitrary interface, AMS Proc. Symp. Appl. Math. W. Gautschi, editor,
48, 311–315.
Li, Z. L.,
& Mikayelyan, H. (2016). Fine numerical analysis of the crack-tip position
for a Mumford-Shah minimizer. Interfaces
and Free Boundaries, 18(1), 75-90.
Li, Z., Qiao,
Z., & Tang, T. (2018). Numerical
solution of differential equations: Introduction to finite difference and
finite element methods. New York: Cambridge University Press.
Li, Z. L.,
& Qin, F. F. (2017). An augmented method for 4th order PDEs with
discontinuous coefficients. Journal of
Scientific Computing, 73(2-3), 968-979.
Li, Z. L.,
& Wang, C. (2003), A fast ?nite difference method for solving Navier-Stokes
equations on irregular domains, J. of
Commu. in Math. Sci., 1, 180–196.
Li, Z. L.,
Xiao, L., Cai, Q., Zhao, H. K., & Luo, R. (2015). A semi-implicit augmented
IIM for Navier-Stokes equations with open, traction, or free boundary
conditions. Journal of Computational
Physics, 297, 182-193.
Li, Z. L.,
Zhao, H. K., & Gao, H. J. (1999). A numerical study of electro-migration
voiding by evolving level set functions on a fixed Cartesian grid. Journal of Computational Physics,
152(1), 281-304.
Mori, Y.
(2008) Convergence proof of the velocity ?eld for a Stokes ?ow immersed
boundary method, Comm. Pure Appl. Math.
61, 1213-1263.
Peskin, C. & McQueen, D. M. (1995), A general method
for the computer simulation of biological systems interacting with ?uids, Symposia of the Society for Experimental
Biology 49, 265.
Xia,
J. L., Li, Z. L., & Ye, X. (2015). Effective matrix-free preconditioning
for the augmented immersed interface method. Journal of Computational Physics, 303, 295-312.