efficient numerical schemes for viscoplastic avalanches part 2 the 2d case abstract this paper deals with the numerical resolution of a shallow water viscoplastic flow model viscoplastic materials are characterized by the existence of a yield stress below a certain critical threshold in the imposed stress there is no deformation and the material behaves like a rigid solid but when that yield value is exceeded the material flows like a fluid in the context of avalanches it means that after going down a slope the material can stop and its free surface has a non trivial shape as opposed to the case of water newtonian fluid the model involves variational inequalities associated with the yield threshold finite volume schemes are used together with duality methods namely augmented lagrangian and bermudez moreno to discretize the problem to be able to accurately simulate the stopping behavior of the avalanche new schemes need to be designed involving the classical notion of well balancing in the present context it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet dry fronts which are encountered in geophysical geometries here we derive such schemes in 2d as the follow up of the companion paper treating the 1d case numerical tests include in particular a generalized 2d benchmark for bingham codes the bingham couette flow with two non zero boundary conditions on the velocity and a simulation of the avalanche path of taconnaz in chamonix mont blanc to show the usability of these schemes on real topographies from digital elevation models dem

analytical bounds for block approximate factorization methods abstract we derive simple analytical upper bounds on the spectral condition number associated with the preconditioning of large sparse linear systems by the so called modified block incomplete factorizations the results prove to handle a large class of discrete second order elliptic partial differential equations and support the conclusion that sparse approximate block matrix factorizations provide powerful preconditioners

a note on the efficient implementation of implicit methods for odes the use of implicit methods for odes e g implicit runge kutta schemes requires the solution of nonlinear systems of algebraic equations of dimension sm where m is the size of the continuous differential problem to be approximated usually the solution of this system represents the most time consuming section in the implementation of such methods consequently the efficient solution of this section would improve their performance in this paper we propose a new iterative procedure to solve such equations on sequential computers

superconvergence of solution derivatives for the shortley weller difference approximation of poisson s equation part i smoothness problems the finite difference method fdm using the shortley weller approximation can be viewed as a special kind of the finite element methods fems using the piecewise bilinear and linear functions and involving some integration approximation when u c3 s i e u c3 0 s and f c2 s the superconvergence rate o h2 of solution derivatives in discrete h1 norms by the fdm is derived for rectangular difference grids where h is the maximal mesh length of difference grids used and the difference grids are not confined to be quasiuniform comparisons are made on the analysis by the maximum principle and the fem analysis conversions between the fdm and the linear and bilinear fems are discussed and numerical experiments are provided to support superconvergence analysis made

conditioning analysis of positive definite matrices by approximate factorizations abstract the conditioning analysis of positive definite matrices by approximate lu factorizations is usually reduced to that of stieltjes matrices or even to more specific classes of matrices by means of perturbation arguments like spectral equivalence we show in the present work that a wider class which we call almost stieltjes matrices can be used as reference class and that it has decisive advantages for the conditioning analysis of finite element approximations of large multidimensional steady state diffusion problems

optimal order preconditioning of finite difference matrices a new multilevel preconditioner is proposed for the iterative solution of linear systems whose coefficient matrix is a symmetric m matrix arising from the discretization of a second order elliptic pde it is based on a recursive block incomplete factorization of the matrix partitioned in a two by two block form in which the submatrix related to the fine grid nodes is approximated by a modified incomplete milu factorization and the schur complement computed from a diagonal approximation of the same submatrix a general algebraic analysis proves optimal order convergence under mild assumptions related to the quality of the approximations of the schur complement on the one hand and of the fine grid submatrix on the other hand this analysis does not require a red black ordering of the fine grid nodes nor a transformation of the matrices in hierarchical form considering more specifically 5 point finite difference approximations of two dimensional problems we prove that the spectrum of the preconditioned system is contained in the interval 0 46 3 independently of possible large jumps or anisotropy in the pde coefficients as long as the latter are piecewise constant on the coarsest mesh numerical results illustrate the efficiency and the robustness of the proposed method

quantitative phase field modeling of dendritic growth in two and three dimensions we report the results of quantitative phase field simulations of the dendritic crystallization of a pure melt in two and three dimensions these simulations exploit a recently developed thin interface limit of the phase field model a karma and w j rappel phys rev e 53 r3017 1996 which is given here a detailed exposition this limit makes it possible to perform efficient computations with a smaller ratio of capillary length to interface thickness and with an arbitrary interface kinetic coefficient simulations in one and two dimensions are first carried out to test the accuracy of phase field computations performed within this limit dendrite tip velocities and tip shapes are found to be in excellent quantitative agreement with exact numerical benchmarks of solvability theory obtained by a boundary integral method both with and without interface kinetics simulations in three dimensions exploit in addition to the asymptotics a methodology to calculate grid corrections due to the surface tension and kinetic anisotropies they are used to test basic aspects of dendritic growth theory that pertain to the selection of the operating state of the tip and to the three dimensional morphology of needle crystals without sidebranches for small crystalline anisotropy simulated values of ensuremath sigma are slightly larger than solvability theory predictions computed by the boundary integral method assuming an axisymmetric shape and agree relatively well with experiments for succinonitrile given the uncertainty in the measured anisotropy in contrast for large anisotropy simulated ensuremath sigma values are significantly larger than the predicted values this disagreement however does not signal a breakdown of solvability theory it is consistent with the finding that the amplitude of the mathrm cos 4 ensuremath varphi mode which measures the departure of the tip morphology from a shape of revolution increases with anisotropy this departure can therefore influence the tip selection in a way that is not accurately captured by the axisymmetric approximation for large anisotropy finally the tail shape at a distance behind the tip that is large compared to the diffusion length is described by a linear law r ensuremath sim z with a slope dr dz that is nearly equal to the ratio of the two dimensional and three dimensional steady state tip velocities furthermore the evolution of the cross section of a three dimensional needle crystal with increasing distance behind the tip is nearly identical to the evolution of a two dimensional growth shape in time in accord with the current theory of the three dimensional needle crystal shape

superconvergence of the shortley weller approximation for dirichlet problems abstract this paper presents a superconvergence property of the shortley weller s w approximation applied to the poisson type dirichlet problem in a bounded domain r 2 with the boundary this means that if the exact solution belongs to c 3 1 then the approximate solution obtained by the s w formula gives o h 3 accuracy at every grid point whose distance to is o h and o h 2 accuracy at other grid points where h denotes the equal mesh size in x and y directions the similar property holds for the case u c l 2 where l 0 or 1 and 0 1 stands for the holder exponent numerical examples are also given which illustrate our results

on sparse block factorization iterative methods a synthetic formalism is proposed for the description of sparse block factorization iterative methods it is used to develop existence convergence and comparison theorems and to compare block factorization schemes against point factorization schemes

a second order accurate symmetric discretization of the poisson equation on irregular domains in this paper we consider the variable coefficient poisson equation with dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization moreover since our discretization matrix is symmetric it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method in addition we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains then we briefly consider stefan problems

a finite difference discretization method for heat and mass transfer with robin boundary conditions on irregular domains abstract this paper proposes a finite difference discretization method for simulations of heat and mass transfer with robin boundary conditions on irregular domains the level set method is utilized to implicitly capture the irregular evolving interface and the ghost fluid method to address variable discontinuities on the interface special care has been devoted to providing ghost values that are restricted by the robin boundary conditions specifically it is done in two steps 1 calculate the normal derivative in cells adjacent to the interface by reconstructing a linear polynomial system 2 successively extrapolate the normal derivative and the ghost value in the normal direction using a linear partial differential equation approach this method produces second order accurate solutions for both the poisson and heat equations with robin boundary conditions and first order accurate solutions for the stefan problems the solution gradients are of first order accuracy as expected it is easy to implement in three dimensional configurations and can be straightforwardly generalized into higher order variants the method thus represents a promising tool for practical heat and mass transfer problems involving robin boundary conditions

extending bicg and bicr methods to solve the stein tensor equation abstract the biconjugate gradients bicg and biconjugate residual bicr methods are attractive ways for solving nonsymmetric linear system a x b in this paper the bicg and bicr methods are extended to solve the high order stein tensor equation the convergent properties of the developed iterative algorithms are studied numerical examples are provided to confirm the theoretical results which demonstrate that the proposed algorithms are effective and feasible for solving the stein tensor equation

on solution uniqueness of elliptic boundary value problems in this paper we consider the problem of solution uniqueness for the second order elliptic boundary value problem by looking at its finite element or finite difference approximations we derive several equivalent conditions which are simpler and easier than the boundedness of the entries of the inverse matrix given in yamamoto et al t yamamoto s oishi q fang discretization principles for linear two point boundary value problems ii numer funct anal optim 29 2008 213 224 the numerical experiments are provided to support the analysis made strictly speaking the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem and this condition should be checked by solving the corresponding eigenvalue problems an application of the equivalent conditions is that we may discover the uniqueness simultaneously while seeking the approximate solutions of elliptic boundary equations

on the lyapunov and stein equations let l cn n and let h k cn n be hermitian matrices the general inertia theorem gives a complete set of relations between the similarity class of l and the congruence class of h when the lyapunov equation lh hl k is satisfied and k 0 in this paper we give some relations between the similarity class of l and the congruence class of k when the lyapunov equation is satisfied and h 0 we also consider the corresponding problem with the stein equation

local boundary element based a new finite difference representation for poisson equations abstract we present a new finite difference method for solving poisson s equation with the dirichlet boundary condition on a more general type of discretization for given domain based on the local boundary element method the method uses the piecewise linear approximation and produce a sparse linear system despite the use of boundary elements the discrete maximum principal is established without any angle condition for the discrete cells of the discretization the convergence behavior is comparable to that of standard finite difference methods on rectangle grids and equally super convergence property is attained on more general meshes when the solution u is in the function class c 2 c 3 0 1 also if u c 3 1 the standard o h 2 convergence is obtained numerical tests are given which illustrate our results

a general incomplete block matrix factorization method abstract a general method for incomplete factorization of m matrices in block matrix form is presented the method generalizes previous methods for pointwise incomplete factorization methods of varga 1960 of wo nicki 1973 and of meijerink and van der vorst 1977 as well as a method for tridiagonal block matrices also we present a new and more general result on the existence of so called modified incomplete factorizations which satisfy a generalized rowsum criterion

high order positivity preserving discontinuous galerkin methods for radiative transfer equations the positivity preserving property is an important and challenging issue for the numerical solution of radiative transfer equations in the past few decades different numerical techniques have been proposed to guarantee positivity of the radiative intensity in several schemes however it is difficult to maintain both high order accuracy and positivity the discontinuous galerkin dg finite element method is a high order numerical method which is widely used to solve the neutron photon transfer equations due to its distinguished advantages such as high order accuracy geometric flexibility suitability for h and p adaptivity parallel efficiency and a good theoretical foundation for stability and error estimates in this paper we construct arbitrarily high order accurate dg schemes which preserve positivity of the radiative intensity in the simulation of both steady and unsteady radiative transfer equations in one and two dimensional geometry by using a combined technique of the scaling positivi

polynomial acceleration of iterative schemes associated with subproper splittings a subproper splitting of a matrix a is a decomposition a b c such that the kernel of a includes that of b while the range of b includes that of a our purpose in the present work is to extend the convergence analysis of polynomial acceleration to the case of iterative schemes associated with subproper splittings in the case of hermitian matrices and consistent systems briefly stated our conclusions show that the regular theory extends to the subproper case provided that convergence to the solution of ax b is understood as convergence to a solution of ax b while b 1 a is understood as b a 0 where b is the moore penrose inverse of b

comparison of eigenvalue ratios in artificial boundary perturbation and jacobi preconditioning for solving poisson equation abstract the shortley weller method is a standard finite difference method for solving the poisson equation with dirichlet boundary condition unless the domain is rectangular the method meets an inevitable problem that some of the neighboring nodes may be outside the domain in this case an usual treatment is to extrapolate the function values at outside nodes by quadratic polynomial the extrapolation may become unstable in the sense that some of the extrapolation coefficients increase rapidly when the grid nodes are getting closer to the boundary a practical remedy which we call artificial perturbation is to treat grid nodes very near the boundary as boundary points the aim of this paper is to reveal the adverse effects of the artificial perturbation on solving the linear system and the convergence of the solution we show that the matrix is nearly symmetric so that the ratio of its minimum and maximum eigenvalues is an important factor in solving the linear system our analysis shows that the artificial perturbation results in a small enhancement of the eigenvalue ratio from o 1 h h m i n to o h 3 and triggers an oscillatory order of convergence instead we suggest using jacobi or ilu type preconditioner on the matrix without applying the artificial perturbation according to our analysis the preconditioning not only reduces the eigenvalue ratio from o 1 h h m i n to o h 2 but also keeps the sharp second order convergence

an efficient milu preconditioning for solving the 2d poisson equation with neumann boundary condition abstract milu preconditioning is known to be the optimal one among all the ilu type preconditionings in solving the poisson equation with dirichlet boundary condition it is optimal in the sense that it reduces the condition number from o h 2 which can be obtained from other ilu type preconditioners to o h 1 however with neumann boundary condition the conventional milu cannot be used since it is not invertible and some milu preconditionings achieved the order o h 1 only in rectangular domains in this article we consider a standard finite volume method for solving the poisson equation with neumann boundary condition in general smooth domains and introduce a new and efficient milu preconditioning for the method in two dimensional general smooth domains our new milu preconditioning achieved the order o h 1 in all our empirical tests in addition in a circular domain with a fine grid the cg method preconditioned with the proposed milu runs about two times faster than the cg with ilu

superconvergent second order cartesian method for solving free boundary problem for invadopodia formation in this paper we present a superconvergent second order cartesian method to solve a free boundary problem with two harmonic phases coupled through the moving interface the model recently proposed by the authors and colleagues describes the formation of cell protrusions the moving interface is described by a level set function and is advected at the velocity given by the gradient of the inner phase the finite differences method proposed in this paper consists of a new stabilized ghost fluid method and second order discretizations for the laplace operator with the boundary conditions dirichlet neumann or robin conditions interestingly the method to solve the harmonic subproblems is superconvergent on two levels in the sense that the first and second order derivatives of the numerical solutions are obtained with the second order of accuracy similarly to the solution itself we exhibit numerical criteria on the data accuracy to get such properties and numerical simulations corroborate these criteria in addition to these properties we propose an appropriate extension of the velocity of the level set to avoid any loss of consistency and to obtain the second order of accuracy of the complete free boundary problem interestingly we highlight the transmission of the superconvergent properties for the static subproblems and their preservation by the dynamical scheme our method is also well suited for quasistatic hele shaw like or muskat like problems

superconvergence analysis of approximate boundary flux calculations certain projection post processing techniques have been proposed for computing the boundary flux for two dimensional problems e g see carey et al 5 in a series of numerical experiments on elliptic problems they observed that these post processing formulas for approximate fluxes were almost o h 2 accurate for linear triangular elements in this paper we prove that the computed boundary flux iso h 2 ln 1 h accurate in the maximum norm for the partial method of 5 if the solutionu h 3 then the boundary flux error iso h 3 2 in thel 2 norm

inertia of the stein transformation with respect to some nonderogatory matrices let a be an n by n nonderogatory matrix all of whose eigenvalues lie on the unit circle and let and be nonnegative integers with n let and be positive integers and a nonnegative integer with n in this paper we explore the existence of a hermitian nonsingular matrix k with inertia 0 such that the stein transformation of k corresponding to a sa k k aka is a hermitian matrix with inertia the study is done by reducing a to jordan canonical form if c is an n by n nonderogatory matrix all of whose eigenvalues lie on the imaginary axis then the results obtained for sa k are valid for the lyapunov transformation lc k ck kc of k corresponding to c

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

possible inertia combinations in the stein and lyapunov equations abstract in the stein or equivalently the lyapunov equation we show that the only joint constraints on the inertias of the three matrices are the classical definiteness and semidefiniteness constraints and a simple rank related constraint

dynamically relaxed block incomplete factorizations for solving two and three dimensional problems to efficiently solve second order discrete elliptic pdes by krylov subspace like methods one needs to use some robust preconditioning techniques relaxed incomplete factorizations rilu are powerful candidates unfortunately their efficiency critically depends on the choice of the relaxation parameter omega whose optimal value is not only hard to estimate but also strongly varies from one problem to another these methods interpolate between the popular ilu and its modified variant milu concerning the pointwise schemes a new variant of rilu that dynamically computes variable omega omega i has been introduced recently like its ancestor rilu and unlike standard methods it is robust with respect to both existence and performance on top of that it breaks the problem dependence of omega opt a block version of this dynamically relaxed method is proposed and compared with classical pointwise and blockwise methods as well as with some existing dynamic variants showing that with the new blockwise preconditioning technique anisotropies are handled more effectively

computation of solid liquid phase fronts in the sharp interface limit on fixed grids a finite difference formulation is applied to track solid liquid boundaries on a fixed underlying grid the interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked the imposition of boundary conditions exactly on a sharp interface that passes through the cartesian grid is performed using simple stencil readjustments in the vicinity of the interface attention is paid to formulating difference schemes that are globally second order accurate in x and t error analysis and grid refinement studies are performed for test problems involving the diffusion and convection diffusion equations and for stable solidification problems issues concerned with stability and change of phase of grid points in the evolution of solid liquid phase fronts are also addressed it is demonstrated that the field calculation is second order accurate while the position of the phase front is calculated to first order accuracy furthermore the accuracy estimates hold for the cases where there is a property jump across the interface unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work the results indicate the need to begin an effort to benchmark computations of instability phenomena

a new finite difference representation for poisson s equation on r3 from a contour integral abstract in this paper we propose a new finite difference representation for solving a dirichlet problem of poisson s equation on r 3 the key idea of the new approach is to represent the solution with a contour integral connecting the nodal values of each local domain centered at each isolated grid node which is based on the boundary integral equation on the local domain and calculate the contour integral using a piecewise linear interpolation of the solution a superconvergence of the scheme is analyzed using a maximum principle and a priori estimate for the finite difference operator the convergence behavior is comparable to that of standard finite difference methods on rectangle grids and a superconvergence property is attained when the solution u is in the function class c 2 c 3 0 u c 3 1 the standard o h 2 convergence is obtained

on the conditioning analysis of block approximate factorization methods abstract the paper is devoted to the conditioning analysis of modified block incomplete factorizations of a given stieltjes matrix we obtain new results improve other theories and compare all existing upper bounds through numerical experiments applied to discrete elliptic pdes our results show that an o h 1 spectral bound can be achieved for a large class of problems

taking advantage of the potentialities of dynamically modified block incomplete factorizations recently modified block incomplete factorizations with dynamic diagonal perturbations have been introduced as preconditioning techniques to solve large linear systems and were successfully tested on isotropic and moderately anisotropic two dimensional partial differential equations pdes an analytic study is performed on the basis of improved versions of results published elsewhere displaying why dynamic methods should be preferred to standard block methods in particular the empirical observation that the optimal choice of the involved parameter does not significantly vary from one problem to another is theoretically confirmed it is also explained why in the case of strong anisotropy lack of attention when adding diagonal perturbations may result in very poor performance rules to surmount this inconvenience are discussed and tested on discretized two dimensional pdes

geometric discretization of the multidimensional dirac delta distribution application to the poisson equation with singular source terms abstract we present a discretization method for the multidimensional dirac distribution we show its applicability in the context of integration problems and for discretizing dirac distributed source terms in poisson equations with constant or variable diffusion coefficients the discretization is cell based and can thus be applied in a straightforward fashion to quadtree octree grids the method produces second order accurate results for integration superlinear convergence is observed when it is used to model dirac distributed source terms in poisson equations the observed order of convergence is 2 or slightly smaller the method is consistent with the discretization of dirac delta distribution for codimension one surfaces presented in 1 2 we present quadtree octree construction procedures to preserve convergence and present various numerical examples including multi scale problems that are intractable with uniform grids

multiscale finite difference diffusion monte carlo method for simulating dendritic solidification abstract we present a novel hybrid computational method to simulate accurately dendritic solidification in the low undercooling limit where the dendrite tip radius is one or more orders of magnitude smaller than the characteristic spatial scale of variation of the surrounding thermal or solutal diffusion field the first key feature of this method is an efficient multiscale diffusion monte carlo dmc algorithm which allows off lattice random walkers to take longer and concomitantly rarer steps with increasing distance away from the solid liquid interface as a result the computational cost of evolving the large scale diffusion field becomes insignificant when compared to that of calculating the interface evolution the second key feature is that random walks are only permitted outside of a thin liquid layer surrounding the interface inside this layer and in the solid the diffusion equation is solved using a standard finite difference algorithm that is interfaced with the dmc algorithm using the local conservation law for the diffusing quantity here we combine this algorithm with a previously developed phase field formulation of the interface dynamics and demonstrate that it can accurately simulate three dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance

effective condition number for finite difference method for solving the linear algebraic equations ax b with the symmetric and positive definite matrix a from elliptic equations the traditional condition number in the 2 norm is defined by cond l n where l and n are the maximal and minimal eigenvalues of the matrix a respectively the condition number is used to provide the bounds of the relative errors from the perturbation of both a and b such a cond can only be reached by the worst situation of all rounding errors and all b for the given b the true relative errors may be smaller or even much smaller than the cond which is called the effective condition number in chan and foulser effectively well conditioned linear systems siam j sci statist comput 9 1988 963 969 and christiansen and hansen the effective condition number applied to error analysis of certain boundary collocation methods j comput appl math 54 1 1994 15 36 in this paper we propose the new computational formulas for effective condition number cond eff and define the new simplified effective condition number cond e for the latter we only need the eigenvector corresponding to the minimal eigenvalue of a which can be easily obtained by the inverse power method in this paper we also apply the effective condition number for the finite difference method for poisson s equation the difference grids are not supposed to be quasiuniform under a non orthogonality assumption the effective condition number is proven to be o 1 for the homogeneous boundary conditions such a result is extraordinary compared with the traditional cond o hmin 2 where hmin is the minimal meshspacing of the difference grids used for the non homogeneous neumann and dirichlet boundary conditions the effective condition number is proven to be o h 1 2 and o h 1 2hmin 1 respectively where h is the maximal meshspacing of the difference grids numerical experiments are carried out to verify the analysis made

high order positivity preserving discontinuous galerkin schemes for radiative transfer equations on triangular meshes abstract it is an important and challenging issue for the numerical solution of radiative transfer equations to maintain both high order accuracy and positivity for the two dimensional radiative transfer equations ling et al give a counterexample d ling j cheng and c w shu j sci comput 77 2018 1801 1831 showing that unmodulated discontinuous galerkin dg solver based either on the p k or q k polynomial spaces could generate negative cell averages even if the inflow boundary value and the source term are both positive and for time dependent problems also a nonnegative initial condition therefore the positivity preserving frameworks in x zhang and c w shu j comput phys 229 2010 8918 8934 and x zhang y xia and c w shu j sci comput 50 2012 29 62 which are based on the value of cell averages being positive cannot be directly used to obtain a high order conservative positivity preserving dg scheme for the radiative transfer equations neither on rectangular meshes nor on triangular meshes in d yuan j cheng and c w shu siam j sci comput 38 2016 a2987 a3019 when the cell average of dg schemes is negative a rotational positivity preserving limiter is constructed which could keep high order accuracy and positivity in the one dimensional radiative transfer equations with p k polynomials and could be straightforwardly extended to two dimensional radiative transfer equations on rectangular meshes with q k polynomials tensor product polynomials this paper presents an extension of the idea of the above mentioned one dimensional rotational positivity preserving limiter algorithm to two dimensional high order positivity preserving dg schemes for solving steady and unsteady radiative transfer equations on triangular meshes with p k polynomials the extension of this method is conceptually plausible but highly nontrivial we first focus on finding a special quadrature rule on a triangle which should satisfy some conditions the most important one is that the quadrature points can be arranged on several line segments on which we can use the one dimensional rotational positivity preserving limiter since the number of the quadrature points is larger than the number of basis functions of p k polynomial space we determine a k th polynomial by a l 2 norm least square subject to its cell average being equal to the weighted average of the values on the quadrature points after using the rotational positivity preserving limiter since the weights used here are the quadrature weights which are positive then the cell average of the modified polynomial is nonnegative and the final modified polynomial can be obtained by using the two dimensional scaling positivity preserving limiter on the triangular element we theoretically prove that our rotational positivity preserving limiter on triangular meshes could keep both high order accuracy and positivity it is relatively simple to implement and also does not affect convergence to weak solutions the numerical results validate the high order accuracy and the positivity preserving property of our schemes the advantage of the triangular meshes on handling complex domain is also presented in our numerical examples

on axelsson s perturbations abstract a nice perturbation technique was introduced by axelsson and further developed by gustafsson to prove that factorization iterative methods are able under appropriate conditions to reach a convergence rate larger by an order of magnitude than that of classical schemes gustafsson observed however that the perturbations introduced to prove this result seemed actually unnecessary to reach it in practice in the present work on the basis of eigenvalue bounds recently obtained by the author we offer an alternative approach which brings a partial confirmation of gustafsson s conjecture

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

lower eigenvalue bounds for singular pencils of matrices abstract beauwens procedure for obtaining lower eigenvalue bounds for regular pencils of matrices a b is simplified and extended to the singular case the theory is then compared through a particular perturbed modified incomplete factorization with notay s generalization of another approach initiated by gustafsson and developed by axelsson and barker and by wilmet

analyses on the finite difference method by gibou et al for poisson equation gibou et al in 4 introduced a finite difference method for solving the poisson equation in irregular domains with the dirichlet boundary condition contrary to its great importance its properties have not been mathematically analyzed but have just been numerically observed in this article we present two analyses for the method one proves that its solution is second order accurate and the other estimates the condition number of its linear system according to our estimation the condition number of the unpreconditioned linear system is of size o 1 h h min and each of jacobi sgs and ilu preconditioned systems is of size o h 2 furthermore our analysis shows that the condition number of milu is of size o h 1 the most successful one

conditioning analysis of modified block incomplete factorizations abstract we improve the conditioning analysis of modified block incomplete factorizations of stieltjes matrices letting n denote the number of diagonal blocks our results show that the spectral condition number is bounded by n for a large class of two dimensional pdes

penalty combinations of the ritz galerkin and finite difference methods for singularity problems abstract penalty combination of the ritz galerkin and finite difference methods is presented for solving elliptic boundary value problems with singularities the superconvergence rate o h2 of solution derivatives by the combination can be achieved while using quasiuniform rectangular difference grids where h is the maximal mesh length of difference grids used in the finite difference method and 0 is an arbitrarily small number it is due to its simplicity that the penalty combination of the ritz galerkin and finite difference methods is highly recommended for solving the complicated problems with multiple singularities and multiple interfaces

on approximate factorization methods for block matrices suitable for vector and parallel processors abstract some existence results for methods based on the approximate factorization of block matrices are proven these methods are based on recursive computations of diagonal block matrices and the approximation of their inverses to preserve sparsity we also discuss a recently proposed 1 inverse free factorization method and present some numerical tests for it

a non negative moment preserving spatial discretization scheme for the linearized boltzmann transport equation in 1 d and 2 d cartesian geometries we present a new nonlinear spatial finite element method for the linearized boltzmann transport equation with s n angular discretization in 1 d and 2 d cartesian geometries this method has two central characteristics first it is equivalent to the linear discontinuous ld galerkin method whenever that method yields a strictly non negative solution second it always satisfies both the zeroth and first spatial moment equations because it yields the ld solution when that solution is non negative one might interpret our method as a classical fix up to the ld scheme however fix up schemes for the ld equations derived in the past have given up solution of the first moment equations when the ld solution is negative in order to satisfy positivity in a simple manner we present computational results comparing our method in 1 d to the strictly non negative linear exponential discontinuous method and to the ld method we present computational results in 2 d comparing our method to a recently developed ld fix up scheme and to the ld scheme it is demonstrated that our method is a valuable alternative to existing methods

conditioning analysis of block incomplete factorizations and its application to elliptic equations the paper deals with eigenvalue estimates for block incomplete fac torization methods for symmetric matrices first some previous results on upper bounds for the maximum eigenvalue of preconditioned matrices are generalized to each eigenvalue second upper bounds for the maximum eigenvalue of the preconditioned matrix are further estimated which presents a substantial im provement of earlier results finally the results are used to estimate bounds for every eigenvalue of the preconditioned matrices in particular for the maximum eigenvalue when a modified block incomplete factorization is used to solve an elliptic equation with variable coefficients in two dimensions the analysis yields a new upper bound of type h 1 for the condition number of the preconditioned matrix and shows clearly how the coefficients of the differential equation influ ence the positive constant

existence and conditioning properties of sparse approximate block factorizations a particular class of sparse approximate block lu factorizations is investigated sufficient conditions are determined for the existence of b lp 1 u for a given coefficient matrix a bounds on the spectral condition number of b 1 a are obtained for symmetric b and a when applied to discrete multidimensional elliptic partial differential equations these results lead under reasonable assumptions to o h 1 spectral condition numbers

the inertia of a hermitian matrix having prescribed complementary principal submatrices abstract for i 1 2 let hi be a given ni ni hermitian matrix we characterize the set of inertias in h 1 x x h 2 x is n 1 n 2 in terms of in h1 and in h2

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

lower eigenvalue bounds for pencils of matrices abstract eigenvalue bounds are obtained for pencils of matrices a vb where a is a stieltjes matrix and b is positive definite under assumptions suitable for the estimation of asymptotic convergence rates of factorization iterative methods where b represents the approximate factorization of a the upper bounds obtained depend on the connectivity structure of the matrices involved which enters through matrix graph considerations in addition a more classical argument is used to obtain a lower bound potential applications of these results include a partial confirmation of gustafsson s conjecture concerning the nonnecessity of axelsson s perturbations

modified block approximate factorization strategies two variants of modified incomplete block matrix factorization with additive correction are proposed for the iterative solution of large linear systems of equations both rigorous theoretical support and numerical evidence are given displaying their efficiency when applied to discrete second order partial differential equations pdes even in the case of quasi singular problems

analysis of a one dimensional model for the immersed boundary method numerical methods are studied for the one dimensional heat equation with a singular forcing term u t u xx c t delta x alpha t the delta function delta x is replaced by a discrete approximation d h x and the resulting equation is solved by a crank nicolson method on a uniform grid the accuracy of this method is analyzed for various choices of d h the case where c t is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied these problems serve as a model for the immersed boundary method of peskin for incompressible flow problems in irregular regions some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions

on a global superconvergence of the gradient of linear triangular elements abstract we study a simple superconvergent scheme which recovers the gradient when solving a second order elliptic problem in the plane by the usual linear elements the recovered gradient globally approximates the true gradient even by one order of accuracy higher in the l 2 norm than the piecewise constant gradient of the ritz galerkin solution a superconvergent approximation to the boundary flux is presented as well

a boundary condition capturing method for incompressible flame discontinuities abstract in this paper we propose a new numerical method for treating two phase incompressible flow where one phase is being converted into the other e g the vaporization of liquid water we consider this numerical method in the context of treating discontinuously thin flame fronts for incompressible flow this method was designed as an extension of the ghost fluid method 1999 j comput phys 152 457 and relies heavily on the boundary condition capturing technology developed in liu et al 2000 j comput phys 154 15 for the variable coefficient poisson equation and in kang et al in press j comput phys for multiphase incompressible flow our new numerical method admits a sharp interface representation similar to the method proposed in helenbrook et al 1999 j comput phys 148 366 since the interface boundary conditions are handled in a simple and straightforward fashion the code is very robust e g no special treatment is required to treat the merging of flame fronts the method is presented in three spatial dimensions with numerical examples in one two and three spatial dimensions

computation of three dimensional dendrites with finite elements starting from an initial seed crystal inside an undercooled liquid the solid phase begins to grow rapidly and develops unstable growth patterns some growth directions are preferred because of anisotropic parameters in the physical model this results in the development of dendrites the physical model includes the heat equation for both the liquid and solid phases the gibbs thomson law couples velocity and curvature of the interface and the temperature we describe a numerical method that enables us to compute dendritic growth of crystals in two and three space dimensions the method consists of two coupled finite element algorithms the first one solves the heat equation the other operates on a discretization of the free boundary and computes the evolution of this moving interface the two methods work with totally independent grids by using timedependent locally refined and coarsened adaptive meshes in both methods we are able to reach a spatial resolution necessary to compute dendritic growth in two and three space dimensions

a survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices eigenvalue and condition number estimates for preconditioned iteration matrices provide the information required to estimate the rate of convergence of iterative methods such as preconditioned conjugate gradient methods in recent years various estimates have been derived for perturbed modified block incomplete factorizations we survey and extend some of these and derive new estimates in particular we derive upper and lower estimates of individual eigenvalues and of condition number this includes a discussion that the condition number of preconditioned second order elliptic difference matrices is o h 1 some of the methods are applied to compute certain parameters involved in the computation of the preconditioner

superconvergence phenomenon in the finite element method arising from averaging gradients we study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements the averaged gradient is a piecewise linear continuous vector field the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point the convergence rate of the averaged gradient to an exact gradient in thel 2 norm can locally be higher even by one than that of the original piecewise constant discrete gradient