The Immersed Interface Method

Transcription

1 The Immersed Interface Method Numerical Solutions of PDEs Involving Interfaces and Irregular Domains Zhiiin Li Kazufumi Ito North Carolina State University Raleigh, North Carolina Society for Industrial and Applied Mathematics Philadelphia

2 Preface xv Introduction A one-dimensional model problem A two-dimensional example of heat propagation in a heterogeneous material Examples of irregular domains and free boundary problems The scope of the monograph and the methodology Jump conditions The choice of grids A minireview of some popular finite difference methods for interface problems The smoothing method for discontinuous coefficients The harmonic averaging for discontinuous coefficients Peskin's immersed boundary (IB) method Numerical methods based on integral equations The ghost fluid method Finite difference and finite volume methods Conventions and notation Cartesian grids Limiting values and jump conditions The local coordinates Interface representations What is the 1IM? 20 The IIM for One-Dimensional Elliptic Interface Problems Reformulating the problem using the jump conditions The IIM for the simple one-dimensional model equation The derivation of the finite difference scheme at an irregular grid point The IIM for general one-dimensional elliptic interface problems The error analysis of the IIM for one-dimensional interface problems One-dimensional numerical examples and a comparison with other methods 30 IX

3 The IIM for Two-Dimensional Elliptic Interface Problems Interface relations for two-dimensional elliptic interface problems The finite difference scheme of the IIM in two dimensions The 6-point finite difference stencil at irregular grid points The fast Poisson solver for problems with only singular sources Enforcing the discrete maximum principle Choosing the finite difference stencil Solving the optimization problem The error analysis of the maximum principle preserving scheme Existence of the solution to the optimization problem The proof of the convergence of the finite difference scheme Some numerical examples for two-dimensional elliptic interface problems Algorithm efficiency analysis Multigrid solvers for large jump ratios 53 The IIM for Three-Dimensional Elliptic Interface Problems A local coordinate system in three dimensions Interface relations for three-dimensional elliptic interface problems The finite difference scheme of the IIM in three dimensions Finite difference equations at regular grid points Computing the orthogonal projection in a three-dimensional Cartesian grid Setting up a local coordinate system using a level set function The bilinear interpolation in three dimensions Deriving the finite difference equation at an irregular grid point Computing surface derivatives of interface quantities in three dimensions The 10-point finite difference stencil at irregular grid points The maximum principle preserving scheme in three dimensions Solving the finite difference equations using an AMG solver A numerical example for a three-dimensional elliptic interface problem 71 Removing Source Singularities for Certain Interface Problems Eliminating source singularities using level set functions: The Theory The finite difference scheme using the new formulation The extension of jump conditions along the normal lines 75

4 xi The orthogonal projections in Cartesian and polar coordinates in two dimensions The discretization strategy using the transformation An outline of the algorithm of removing source singularities A closed formula for the correction terms Computing the gradient using the new formulation An example of removing source singularities Removing source singularities for variable coefficients Orthogonal projections and extensions in spherical coordinates Augmented Strategies The augmented technique for elliptic interface problems The augmented variable for the elliptic interface problems 90.2 The discrete system of equations in matrix-vector form The least squares interpolation scheme from a Cartesian grid to an interface 94.4 Invertibility of the Schur complement system 97.5 A preconditioner for the Schur complement system 98.6 Numerical experiments and analysis of the fast IIM The augmented method for generalized Helmholtz equations on irregular domains An example of the augmented approach for Poisson equations on irregular domains The Fourth-Order IIM Two-point boundary value problems The constant coefficient case Ill General boundary conditions Ill The smooth variable coefficient case The piecewise constant coefficient case Two-dimensional cases The fourth-order compact central finite difference method Neumann boundary conditions The fourth-order method for Poisson equations on irregular domains Projections and a fourth-order polynomial interpolation The fourth-order method for heat equations on irregular domains The fourth-order method for PDEs with variable coefficient on irregular domains 127

5 xii Contents The fourth-order method for interface problems The fourth-order method for heat equations with interfaces The fourth-order methods for three dimensional cases The fourth-order scheme for problems on irregular domains in three dimensions The fourth-order scheme for three-dimensional interface problems The preconditioned subspace iteration method The irregular domain case The interface case Numerical experiments The irregular domain case Examples for eigenvalues and eigenfunctions in a circular domain Results for the variable coefficient case Results for the interface problem An eigenvalue problem with an interface The well-posedness and the convergence rate Convergence rate The Immersed Finite Element Methods The IFEM for one-dimensional interface problems New basis functions satisfying the jump conditions The interpolation functions in the one-dimensional IFEM space The convergence analysis for the one-dimensional IFEM A numerical example of one-dimensional IFEM The weak form of two-dimensional elliptic interface problems A nonconforming IFE space and analysis Local basis functions on an interface element The nonconforming IFE space Approximation properties of the nonconforming IFE space A nonconforming IFEM A conforming IFE space and analysis The conforming local basis functions on an interface element A conforming IFE space Approximation properties of the conforming IFE space A numerical example and analysis for IFEMs Numerical results for the conforming IFEM A comparison with the finite element method with added nodes IFEM for problems with nonhomogeneous jump conditions 186

6 xiii 9 The IIM for Parabolic Interface Problems The IIM for one-dimensional heat equations with fixed interfaces The IIM for one-dimensional moving interface problems The modified Crank-Nicholson scheme Dealing with grid crossing The discretizations of u x and (fiu x ) x near the interface Computing interface quantities "' Solving the resulting nonlinear system of equations Validation of the algorithm for a one-dimensional moving interface problem The modified ADI method for heat equations with discontinuities The modified ADI scheme Determining the spatial correction terms Decomposing the jump condition in the coordinate directions The local truncation error analysis for the ADI method A numerical example of the modified ADI method The IIM for diffusion and advection equations Determining the finite difference coefficients for the diffusion term Determining the finite difference coefficients for the advection term The IIM for Stokes and Navier-Stokes Equations The derivation of the jump conditions for Stokes and Navier-Stokes equations The IIM for Stokes equations with singular sources: The membrane model The force density of the elastic membrane model Solving the Poisson equation for the pressure Solving the Poisson equations for the velocity (u,v) Evolving the interface using an explicit method Evolving the interface using an implicit method The validation of the IIM for moving elastic membranes The IIM for Stokes equations with singular sources: The surface tension model An augmented approach for Stokes equations with discontinuous viscosity The augmented algorithm for Stokes equations The validation of the augmented method for Stokes equations An augmented approach for pressure boundary conditions Computing the Laplacian of the velocity along a boundary for a nonslip boundary condition 249

7 xiv Contents 10.6 The IIM for Navier-Stokes equations with singular sources Additional interface relations The modified finite difference method for Navier-Stokes equations with interfaces Determining the correction terms Correction terms to the projection method Further corrections near the boundary and the interface " Comparisons and validation of the IIM for Navier-Stokes equations with interfaces Some Applications of the IIM The framework coupling the IIM with evolution schemes The front-tracking method Coupling the level set method with the IIM Orthogonal projections and the bilinear interpolation Velocity extension along normal directions Reconstructing the interface locally from a level set function The hybrid IIM-level set method for the Hele-Shaw flow Dynamic stability of the Hele-Shaw flow The IIM for the Hele-Shaw flow Numerical experiments of the Hele-Shaw flow Simulations of Stefan problems and crystal growth A modified Crank-Nicolson discretization The modified ADI method for Stefan problems Numerical simulations of the Stefan problem An application to an inverse problem of shape identification An outline of the algorithm for the inverse problem Identifying several minima Numerical examples of shape identification Applications to nonlinear interface problems The substitution method Computing p and its derivatives Numerical experiments of MR fluids with particles Other methods related to the IIM The IIM for hyperbolic systems of PDEs The explicit jump immersed interface method (EJIIM) The high-order matched interface and boundary method Future directions 309 Bibliography 311 Index 331