discrete singular convolution and its application to the analysis of plates with internal supports part 2 applications part 2 of this series of two papers presents the applications of the discrete singular convolution dsc algorithm the main purpose of this paper is to explore the utility test the accuracy and examine the convergence of the proposed approach for the vibration analysis of rectangular plates with internal supports both partial internal line supports and complex internal supports are considered for 21 square plates of various combinations of edge support conditions the effects of different size shape and topology of the internal supports and different boundary conditions on the vibration response of plates are investigated the partial internal line supports may vary from a central point support to a full range of cross or diagonal line supports several closed loop supports such as ring square and rhombus and their combinations are studied for complex internal supports convergence and comparison studies are carried out to establish the correctness and accuracy of the dsc algorithm the dsc results are compared with those in the available literature obtained by using other methods numerical results indicate that the dsc algorithm exhibits controllable accuracy for plate analysis and shows excellent flexibility in handling complex geometries boundary conditions and support conditions copyright 2002 john wiley sons ltd

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

high order fdtd methods via derivative matching for maxwell s equations with material interfaces this paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high order finite difference time domain fdtd schemes of computational electromagnetics cem with material interfaces in one 1d and two spatial dimensions 2d by making use of fictitious points systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage to arbitrarily high orders of accuracy in principle while often limited by numerical instability orders up to 16 and 12 are achieved respectively in 1d and 2d detailed stability analyses are presented for the present approach to examine the upper limit in constructing embedded fdtd methods as natural generalizations of the high order fdtd schemes the proposed derivative matching methods automatically reduce to the standard fdtd schemes when the material interfaces are absent an interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code another feature of the present approach is that it can be robustly implemented with other high accuracy time domain approaches such as the multiresolution time domain method and the local spectral time domain method to cope with material interfaces numerical experiments on both 1d and 2d problems are carried out to test the convergence examine the stability access the efficiency and explore the limitation of the proposed methods it is found that operating at their best capacity the proposed high order schemes could be over 2000 times more efficient than their fourth order versions in 2d in conclusion the present work indicates that the proposed hierarchical derivative matching methods might lead to practical high order schemes for numerical solution of time domain maxwell s equations with material interfaces

a unified approach for the solution of the fokker planck equation this paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the fokker planck equation the unified features of the discrete singular convolution algorithm are discussed it is demonstrated that different implementations of the present algorithm such as global local galerkin collocation and finite difference can be deduced from a single starting point three benchmark stochastic systems the repulsive wong process the black scholes equation and a genuine nonlinear model are employed to illustrate the robustness and to test the accuracy of the present approach for the solution of the fokker planck equation via a time dependent method an additional example the incompressible euler equation is used to further validate the present approach for more difficult problems numerical results indicate that the present unified approach is robust and accurate for solving the fokker planck equation

dsc analysis of free edged beams by an iteratively matched boundary method this communication introduces a novel scheme for the treatment of free edge supports in the analysis of beams by using the discrete singular convolution dsc algorithm accommodating free edges has been a challenging issue in the dsc analysis of beams plates and shells an iteratively matched boundary imb method is proposed to overcome the difficulty numerical experiments are carried out to demonstrate that the proposed imb method works very well in dealing with arbitrary combinations of beam edge supports

operator splitting adi schemes for pseudo time coupled nonlinear solvation simulations this work introduces novel operator splitting alternating direction implicit adi schemes to overcome numerical difficulties in solving pseudo time coupled nonlinear partial differential equations pdes for biomolecular solvation analysis based on the variational formulation a pseudo transient continuation model has been previously formulated to couple a nonlinear poisson boltzmann npb equation for the electrostatic potential with a generalized laplace beltrami equation defining the biomolecular surface however the standard numerical integration of the time dependent npb equation is known to be very inefficient moreover it encounters instability issues for smoothly varying solute solvent interfaces so that a filtering process has to be conducted in the present work we propose to solve the unsteady npb equation in an operator splitting framework so that the nonlinear instability can be completely avoided through an analytical integration central finite differences are employed to discretize the nonhomogeneous diffusion term of the generalized npb equation to form tridiagonal matrices in the douglas and douglas rachford type adi schemes the proposed time splitting adi schemes are found to be unconditionally stable for solving the npb equation in benchmark examples with analytical solutions for the solvation analysis involving two pseudo time coupled nonlinear pdes the time stability of the npb equation can be maintained by using very large time increments so that without sacrificing the accuracy the present biomolecular simulation becomes over ten times faster

alternating direction ghost fluid methods for solving the heat equation with interfaces abstract this work presents two new alternating direction implicit adi schemes for solving parabolic interface problems in two and three dimensions first the ghost fluid method gfm is adopted for the first time in the literature to treat interface jumps in the adi framework which results in symmetric and tridiagonal finite difference matrices in each adi step the proposed gfm adi scheme achieves a first order of accuracy in both space and time as confirmed by numerical experiments involving complex interface shapes and spatial temporal dependent jumps the gfm adi scheme also maintains the adi efficiency the computational complexity for each time step scales as o n for a total degree of freedom n in higher dimensions second a new matched interface and boundary mib scheme is constructed which downgrades the quadratic polynomial bases in the existing second order mib to linear ones interestingly the resulting mib adi or madi scheme produces the same finite difference matrices as the gfm adi scheme in all dimensions hence the madi scheme can be regarded as an improvement of the gfm because it calculates tangential jumps which are omitted in the gfm consequently the present madi scheme is constantly more accurate than the gfm adi in all numerical examples while keeping the same computational efficiency nevertheless the madi scheme is semi implicit due to tangential jump approximations while the gfm adi scheme is fully implicit without tangential corrections thus the gfm adi scheme could be more stable than the madi scheme when a huge contrast is presented in diffusion coefficients

a spatially second order alternating direction implicit adi method for solving three dimensional parabolic interface problems abstract a new matched alternating direction implicit adi method is proposed in this paper for solving three dimensional 3d parabolic interface problems with discontinuous jumps and complex interfaces this scheme inherits the merits of its ancestor for two dimensional problems while possesses several novel features such as a non orthogonal local coordinate system for decoupling the jump conditions two side estimation of tangential derivatives at an interface point and a new douglas rachford adi formulation that minimizes the number of perturbation terms to attack more challenging 3d problems in time discretization this new adi method is found to be of first order and stable in various experiments in space discretization the matched adi method achieves the second order accuracy based on simple cartesian grids for various irregularly shaped surfaces and spatial temporal dependent jumps computationally the matched adi method is as efficient as the fastest implicit scheme based on the geometrical multigrid for solving 3d parabolic equations in the sense that its complexity in each time step scales linearly with respect to the spatial degree of freedom n i e o n furthermore unlike iterative methods the adi method is an exact or non iterative algebraic solver which guarantees to stop after a certain number of computations for a fixed n therefore the proposed matched adi method provides a very promising tool for solving 3d parabolic interface problems

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

differential quadrature solutions of eighth order boundary value differential equations special cases of linear eighth order boundary value problems have been solved using polynomial splines however divergent results were obtained at points adjacent to boundary points this paper presents an accurate and general approach to solve this class of problems utilizing the generalized differential quadrature rule gdqr proposed recently by the authors explicit weighting coefficients are formulated to implement the gdqr for eighth order differential equations a mathematically important by product of this paper is that a new kind of hermite interpolation functions is derived explicitly for the first time linear and non linear illustrations are given to show the practical usefulness of the approach developed using frechet derivatives non linear eighth order problems are also solved for the first time numerical results obtained using even only seven sampling points are of excellent accuracy and convergence in an entire domain the present gdqr has shown clear advantages over the existing methods and demonstrated itself as a general stable and accurate numerical method to solve high order differential equations

partitioning rectangular and structurally unsymmetric sparse matrices for parallel processing a common operation in scientific computing is the multiplication of a sparse rectangular or structurally unsymmetric matrix and a vector in many applications the matrix transpose vector product is also required this paper addresses the efficient parallelization of these operations we show that the problem can be expressed in terms of partitioning bipartite graphs we then introduce several algorithms for this partitioning problem and compare their performance on a set of test matrices

optimal error estimates for linear parabolic problems with discontinuous coefficients a finite element discretization is proposed and analyzed for a linear parabolic problems with discontinuous coefficients due to low global regularity of the solution it seems difficult to achieve optimal order of convergence with classical finite element methods numer math 79 1998 pp 175 202 in this paper we have used a finite element discretization where interface triangles are assumed to be curved triangles instead of straight triangles as in classical finite element methods optimal order error estimates in l2 and h1 norms are shown to hold even if the regularity of the solution is low on the whole domain while the continuous time galerkin method is discussed for the spatially discrete scheme the discontinuous galerkin method is analyzed for the fully discrete scheme the interfaces and boundaries of the domains are assumed to be smooth for our purpose

matched interface and boundary mib for the implementation of boundary conditions in high order central finite differences high order central finite difference schemes encounter great difficulties in implementing complex boundary conditions this paper introduces the matched interface and boundary mib method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high order central finite difference schemes to attain arbitrarily high order the mib method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions the proposed approach is extensively validated via boundary value problems initial boundary value problems eigenvalue problems and high order differential equations successful implementations are given to not only dirichlet neumann and robin boundary conditions but also more general ones such as multiple boundary conditions in high order differential equations and time dependent boundary conditions in evolution equations detailed stability analysis of the mib method is carried out the mib method is shown to be able to deliver high order accuracy while maintaining the same or similar stability conditions of the standard high order central difference approximations the application of the proposed mib method to the boundary treatment of other non standard high order methods is also considered

matched interface and boundary mib method for elliptic problems with sharp edged interfaces elliptic problems with sharp edged interfaces thin layered interfaces and interfaces that intersect with geometric boundary are notoriously challenging to existing numerical methods particularly when the solution is highly oscillatory this work generalizes the matched interface and boundary mib method previously designed for solving elliptic problems with curved interfaces to the aforementioned problems we classify these problems into five distinct topological relations involving the interfaces and the cartesian mesh lines flexible strategies are developed to systematically extends the computational domains near the interface so that the standard central finite difference scheme can be applied without the loss of accuracy fictitious values on the extended domains are determined by enforcing the physical jump conditions on the interface according to the local topology of the irregular point the concepts of primary and secondary fictitious values are introduced to deal with sharp edged interfaces for corner singularity or tip singularity an appropriate polynomial is multiplied to the solution to remove the singularity extensive numerical experiments confirm the designed second order convergence of the proposed method

a matched peaceman rachford adi method for solving parabolic interface problems a new peaceman rachford alternating direction implicit pr adi method is proposed in this work for solving two dimensional 2d parabolic interface problems with discontinuous solutions the classical adi schemes are known to be inaccurate for handling interfaces this motivates the development of a matched douglas adi d adi method in the literature in which the finite difference is locally corrected according to the jump conditions however the unconditional stability of the matched adi method cannot be maintained if the d adi is simply replaced by the pr adi to stabilize the computation the tangential derivative approximations in the jump conditions decomposition are substantially improved in this paper moreover a new temporal discretization is adopted for formulating the pr adi method which involves less perturbation terms stability analysis is conducted through eigenvalue spectrum analysis which demonstrates the unconditional stability of the proposed method the matched pr adi method achieves second order of accuracy in space in all tested problems with complex geometries and jumps while maintaining the efficiency of the adi the proposed pr adi method is found to be more accurate than the d adi method in time integration even though its formal temporal order is limited in the matched adi framework

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

hypergraph partitioning based models and methods for exploiting cache locality in sparse matrix vector multiplication sparse matrix vector multiplication spmxv is a kernel operation widely used in iterative linear solvers the same sparse matrix is multiplied by a dense vector repeatedly in these solvers matric

cache oblivious sparse matrix vector multiplication by using sparse matrix partitioning methods in this article we introduce a cache oblivious method for sparse matrix vector multiplication our method attempts to permute the rows and columns of the input matrix using a recursive hypergraph based sparse matrix partitioning scheme so that the resulting matrix induces cache friendly behavior during sparse matrix vector multiplication matrices are assumed to be stored in row major format by means of the compressed row storage crs or its variants incremental crs and zig zag crs the zig zag crs data structure is shown to fit well with the hypergraph metric used in partitioning sparse matrices for the purpose of parallel computation the separated block diagonal sbd form is shown to be the appropriate matrix structure for cache enhancement we have implemented a run time cache simulation library enabling us to analyze cache behavior for arbitrary matrices and arbitrary cache properties during matrix vector multiplication within a k way set associative idealized cache model the results of these simulations are then verified by actual experiments run on various cache architectures in all these experiments we use the mondriaan sparse matrix partitioner in one dimensional mode the savings in computation time achieved by our matrix reorderings reach up to 50 percent in the case of a large link matrix

classroom note calculation of weights in finite difference formulas the classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope e g specialized recursions for centered and staggered approximations for adams bashforth adams moulton and bdf formulas for odes etc two recent algorithms overcome these problems for equispaced grids such weights can be found very conveniently with a two line algorithm when using a symbolic language such as mathematica reducing to one line in the case of explicit approximations for arbitrarily spaced grids we describe a computationally very inexpensive numerical algorithm

some classification results for hyperbolic equations f x y u ux uy uxx uxy uyy 0 abstract we provide a contact invariant characterization for equations of the form u xy a x y u u x b x y u u y c x y u 0 u xy a x y u x b x y u y c x y u 0 uxy c x y u 0 uxy 0 we classify all equations of the form uxy f x y u ux uy 0 for which the two ovsiannikov s invariants are constants these results include characterization of the klein gordon equation uxy u the liouville equation uxy eu and the class of euler poisson darboux equations it is shown that the wave equation uxy 0 liouville equation and the linear equation uxy 2u x y 2 are the only variational equations darboux integrable at level one we also show that a hyperbolic monge ampere equation darboux integrable at level one is equivalent to an equation of type uxy f x y u ux uy 0 we prove that the hyperbolic fermi ulam pasta fpu equation uyy ux 2 uxx is contact equivalent to a linear equation of type uxy c x y u and we classify all fpu equations darboux integrable at level one we also apply our results to equations of type uxy f u ux that describe pseudo spherical surfaces

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

immersed interface methods for neumann and related problems in two and three dimensions we develop and apply a finite difference method to discretize the laplacian operator with neumann boundary conditions on an irregular domain using a regular cartesian grid the method is an extension of the immersed interface method developed by leveque and li siam j numer anal 31 1994 1019 1044 with careful selection of stencils the method is second order accurate and produces a matrix that is stable diagonally semidominant the method is illustrated on several two dimensional problems and one three dimensional problem

characteristics and the geometry of hyperbolic equations in the plane abstract this paper focuses on the implementation and application of elie cartan s ideas on characteristics for second order non linear hyperbolic equations in the plane the topics include a discussion of both intrinsic and extrinsic contact equivalence a description of the theory of backlund and laplace transformations based on characteristics a geometric approach to the notion of riemann invariants and a study of the various possibilities for their existence resulting in a new simple proof of lie s description of the contact orbit of the wave equation the initial value problem is analyzed from the viewpoint of exterior differential systems and applied to establish a smooth existence theorem with an application to a shock wave problem

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

a space time multigrid method for parabolic partial differential equations we consider the solution of parabolic partial differential equations pdes in standard time stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time step by contrast the method presented in this paper treats the whole of the space time problem simultaneously thus the multigrid operations of smoothing and coarse grid correction are defined on all of the space time variables of a given grid level the method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil numerical results for the one and two dimensional heat equations are presented and are shown to agree closely with predictions from fourier mode analysis

a dimension by dimension splitting immersed interface method for heat conduction equation with interfaces the numerical method proposed in this paper is an improvement of the adi method by li and mayo 1994 the proposed method is unconditionally stable for both two and three dimensional heat conduction interface problems while li s adi method is only stable for two dimensional problems the method is a modification of a locally one dimensional lod difference scheme with correction term added to the right hand side of the standard lod difference scheme at irregular points the correction term is determined so that the local truncation error is of order o h at irregular points then the method is two order convergent in both time and space directions numerical examples show good agreement with exact solutions and confirm the order of convergence and stability

the immersed interface technique for parabolic problems with mixed boundary conditions a finite difference scheme is presented for a parabolic problem with mixed boundary conditions we use an immersed interface technique to discretize the neumann condition and we use the shortley weller approximation for the dirichlet condition the proof of a discrete maximum principle is given as well as the proof of convergence of the scheme this convergence is also validated on numerical examples

high accuracy algorithms for computational aeroacoustics this paper presents an analysis of high bandwidth operators developed for use with an essentially nonoscillatory eno method the spatial operators of a sixth order eno code are modified to resolve waves with as few as 7 points per wavelength ppw by decreasing the formal order of the algorithm numerical and analytical solutions are compared for the model problems of plane wave propagation and sound generation by an oscillating sphere these problems involve linear propagation wave steepening and shock formation an analysis of the ppw required for sufficient accuracy shows that low order algorithms need an excessive number of grid points to produce acceptable solutions in contrast high order codes provide good predictions on relatively coarse grids the high bandwidth operators produce only modest improvements over the original sixth order operators for nonlinear problems in which wave steepening is significant however they clearly outperform the original operators for long distance linear propagation because the high bandwidth operators have the same stencil as the original sixth order operators these gains are achieved with no additional computational work

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

an effective implementation of the lin kernighan traveling salesman heuristic abstract this paper describes an implementation of the lin kernighan heuristic one of the most successful methods for generating optimal or near optimal solutions for the symmetric traveling salesman problem tsp computational tests show that the implementation is highly effective it has found optimal solutions for all solved problem instances we have been able to obtain including a 13 509 city problem the largest non trivial problem instance solved to optimality today

a two dimensional data distribution method for parallel sparse matrix vector multiplication a new method is presented for distributing data in sparse matrix vector multiplication the method is two dimensional tries to minimize the true communication volume and also tries to spread the computation and communication work evenly over the processors the method starts with a recursive bipartitioning of the sparse matrix each time splitting a rectangular matrix into two parts with a nearly equal number of nonzeros the communication volume caused by the split is minimized after the matrix partitioning the input and output vectors are partitioned with the objective of minimizing the maximum communication volume per processor experimental results of our implementation mondriaan for a set of sparse test matrices show a reduction in communication volume compared to one dimensional methods and in general a good balance in the communication work experimental timings of an actual parallel sparse matrix vector multiplication on an sgi origin 3800 computer show that a sufficiently large reduction in communication volume leads to savings in execution time

partitioning hypergraphs in scientific computing applications through vertex separators on graphs the modeling flexibility provided by hypergraphs has drawn a lot of interest from the combinatorial scientific community leading to novel models and algorithms their applications and development of associated tools hypergraphs are now a standard tool in combinatorial scientific computing the modeling flexibility of hypergraphs however comes at a cost algorithms on hypergraphs are inherently more complicated than those on graphs which sometimes translates to nontrivial increases in processing times neither the modeling flexibility of hypergraphs nor the runtime efficiency of graph algorithms can be overlooked therefore the new research thrust should be how to cleverly trade off between the two this work addresses one method for this trade off by solving the hypergraph partitioning problem by finding vertex separators on graphs specifically we investigate how to solve the hypergraph partitioning problem by seeking a vertex separator on its net intersection graph nig where each net of the hypergraph is represented by a vertex and two vertices share an edge if their nets have a common vertex we propose a vertex weighting scheme to attain good node balanced hypergraphs since the nig model cannot preserve node balancing information vertex removal and vertex splitting techniques are described to optimize cut net and connectivity metrics respectively under the recursive bipartitioning paradigm we also developed implementations of our proposed hypergraph partitioning formulations by adopting and modifying a state of the art graph partitioning by vertex separator tool onmetis experiments conducted on a large collection of sparse matrices demonstrate the effectiveness of our proposed techniques

an obstruction to the integrability of a class of non linear wave equations by 1 stable cartan characteristics we examine in detail the cauchy problem for a class of non linear hyperbolic equations in two independent variables this class is motivated by the analysis of the dynamics of a line of non linearly coupled particles by fermi pasta and ulam and extends the recent investigation of this problem by gardner and kamran we find conditions for the existence of a 1 stable cartan characteristic of a pfaffian exterior differential system whose integral curves provide a solution to the cauchy problem the same obstruction to involution is exposed in darboux s method of integration and the two approaches are compared a class of particular solutions to the obstruction is constructed

three dimensional matched interface and boundary mib method for treating geometric singularities this paper reports the three dimensional 3d generalization of our previous 2d higher order matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and non smooth interfaces new mib algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier mib scheme for treating interfaces with sharp geometric singularities such as sharp edges sharp wedges and sharp tips the resulting 3d mib schemes are of second order accuracy for arbitrarily complex interfaces with sharp geometric singularities of fourth order accuracy for complex interfaces with moderate geometric singularities and of sixth order accuracy for curved smooth interfaces a systematical procedure is introduced to make the mib matrix optimally symmetric and banded by appropriately choosing auxiliary grid points consequently the new mib linear algebraic equations can be solved with fewer number of iterations the proposed mib method makes use of cartesian grids standard finite difference schemes lowest order interface jump conditions and fictitious values the interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation while a pair of fictitious values are determined along a mesh at a time an iterative procedure is proposed to determine all the required fictitious values for higher order schemes by repeatedly using the lowest order jump conditions a variety of mib techniques are developed to overcome geometric constraints the essential strategy of the mib method is to locally reduce a 2d or a 3d interface problem into 1d like ones the proposed mib method is extensively validated in terms of the order of accuracy the speed of convergence the number of iterations and cpu time numerical experiments are carried out to complex interfaces including the molecular surfaces of a protein a missile interface and van der waals surfaces of intersecting spheres

an augmented matched interface and boundary mib method for solving elliptic interface problem abstract in this paper a second order accurate augmented matched interface and boundary mib is introduced for solving two dimensional 2d elliptic interface problems with piecewise constant coefficients the augmented mib seamlessly combines several key ingredients of the standard mib augmented immersed interface method iim and explicit jump iim to produce a new fast interface algorithm based on the mib zeroth and first order jump conditions are enforced across an arbitrarily curved interface which yields fictitious values on cartesian nodes near the interface by using such fictitious values a simple procedure is proposed to reconstruct cartesian derivative jumps as auxiliary variables and couple them with the jump corrected taylor series expansions which allow us to restore the order of the central difference across the interface to two moreover by using the schur complement to disassociate the algebraic computation of auxiliary variables and function values the discrete laplacian can be efficiently inverted by using the fast fourier transform fft it is found in our numerical experiments that the iteration number in solving the auxiliary system weakly depends on the mesh size as a consequence the total computational cost of the augmented mib is about o n 2 log n for a cartesian grid with dimension n n in 2d therefore the augmented mib outperforms the classical mib in all cases by significantly reducing the cpu time while keeping the same second order of accuracy in dealing with complicated interfaces

on equations of type uxt f u ux which describe pseudospherical surfaces the equations uxt f u ux which describe pseudospherical surfaces are characterized in particular when f does not depend on ux the sine gordon sinh gordon and liouville equations are essentially obtained moreover it is shown that an equation uxt f u has a self baclund transformation if and only if it describes an pseudospherical surface