stability and convergence of the peaceman rachford adi method for initial boundary value problems etude des proprietes de stabilite et de convergence de la methode des directions alternees implicite losrsqu elle est appliquee aux problemes aux valeurs initiales dans un espace bidimensionnel

localization of solutions of anisotropic parabolic equations abstract we study the localization properties of solutions of the dirichlet problem for the anisotropic parabolic equations u t i 1 n d i a i z u d i u p i 2 d i u f z z x t 0 t with constant exponents p i 1 and x r n n 2 such equations arise from the mathematical description of diffusion processes it is shown that if the equation combines the directions of slow diffusion for which p i 2 and the directions of fast or linear diffusion corresponding to p i 1 2 or p 2 then the solutions may simultaneously display the properties intrinsic for the solutions of isotropic equations of fast or slow diffusion under the assumptions that f 0 for t t f and u 0 0 f 0 for x 1 s we show on the one hand that the solution vanishes in a finite time if n 2 i 1 n 1 p i 1 n 2 and on the other hand that the support of the same solution never reaches the plane x 1 s provided that 1 n 1 1 n 1 i 2 n 1 p i 1 p 1

fast and stable explicit operator splitting methods for phase field models numerical simulations of phase field models require long time computations and therefore it is necessary to develop efficient and highly accurate numerical methods in this paper we propose fast and stable explicit operator splitting methods for both one and two dimensional nonlinear diffusion equations for thin film epitaxy with slope selection and the cahn hilliard equation the equations are split into nonlinear and linear parts the nonlinear part is solved using a method of lines together with an efficient large stability domain explicit ode solver the linear part is solved by a pseudo spectral method which is based on the exact solution and thus has no stability restriction on the time step size we demonstrate the performance of the proposed methods on a number of one and two dimensional numerical examples where different stages of coarsening such as the initial preparation alternating rapid structural transition and slow motion can be clearly observed

translating solitons in riemannian products abstract in this paper we study solitons invariant with respect to the flow generated by a complete parallel vector field in a ambient riemannian manifold a special case occurs when the ambient manifold is the riemannian product r p d t 2 g 0 and the parallel field is x t similarly to what happens in the euclidean setting we call them translating solitons we see that a translating soliton in r p can be seen as a minimal submanifold for a weighted volume functional moreover we show that this kind of solitons appear in a natural way in the context of a monotonicity formula for the mean curvature flow in r p when g 0 is rotationally invariant and its sectional curvature is non positive we are able to characterize all the rotationally invariant translating solitons furthermore we use these families of new examples as barriers to deduce several non existence results

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

an implicit explicit runge kutta chebyshev finite element method for the nonlinear lithium ion battery equations abstract we introduce a numerical method to integrate the nonlinear system of equations that model the physical chemical laws of the lithium ion batteries the mathematical model formulated in doyle et al j electrochem soc 140 1993 1526 1533 is a system of strongly coupled nonlinear parabolic and elliptic equations our numerical method combines a second order implicit explicit runge kutta chebyshev scheme for time discretization of the parabolic equations governing the dynamics of the variables ce and cs with linear finite element space discretization of the system the implicit explicit numerical formulation of the parabolic equations allows us to decouple the strong nonlinear reaction terms which are treated implicitly from the linear diffusion terms treated explicitly this approach is computationally efficient because a it has an extended stability region and b the coupled system of algebraic equations becomes a single variable nonlinear equation per mesh point which is easily solved by newton method

on some modification navier stokes equations abstract this article investigates some modification of the navier stokes equations of type as the modification that was suggested by lions quelques methodes de resolution des problemes aux limites non lineares dunod gauthier villaris paris 1969 in the form u j t i 1 n d i u j p i 2 d i u j i 1 n u i d i u j p x j h j t x t x q r n 1 div u 0 t x q 0 t t 0 0 u j 0 x 0 x u 0 0 t under p i p i 1 i n n 2 in this we prove the existence theorem for the different pi max 2 3 2 n and on some additional conditions i e of the pi p 4 in the isotropic nonlinearity case we prove the uniqueness theorem for the considered problem

order results for implicit runge kutta methods applied to stiff systems an extensive order analysis is given for the local and global errors of certain classes of runge kutta methods applied to general nonlinear stiff initial value problems it turns out that there are three order levels for linear problems y ay the classical orders apply for general nonlinear problems however two types of order reductions occur there a distinction has to be made between stiff problems where only f y is large and stiff problems where also some of the other derivatives f t f tt f yy cdots are large for these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h the order levels for both classes of problems are considerably lower than the respective conventional orders these order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems

the calabi yau conjectures for embedded surfaces in this talk i will discuss the proof of the calabi yau conjectures for embedded surfaces this is joint work with bill minicozzi cm9 the calabi yau conjectures about surfaces date back to the 1960s much work has been done on them over the past four decades in particular examples of jorge xavier from 1980 and nadirashvili from 1996 showed that the immersed versions were false we will show here that for embedded surfaces i e injective immersions they are in fact true their original form was given in 1965 in ca where e calabi made the following two conjectures about minimal surfaces see also s s chern page 212 of ch conjecture 1 prove that a complete minimal hypersurface in r must be unbounded calabi continued it is known that there are no compact minimal submanifolds of r or of any simply connected complete riemannian manifold with sectional curvature 0 a more ambitious conjecture is conjecture 2 a complete minimal hypersurface in r has an unbounded projection in every n 2 dimensional flat subspace

a class of high order runge kutta chebyshev stability polynomials the analytic form of a new class of factorized runge kutta chebyshev frkc stability polynomials of arbitrary order n is presented roots of frkc stability polynomials of degree l m n are used to construct explicit schemes comprising l forward euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to l 2 the associated stability domain scales as m 2 along the real axis marginally stable real valued points on the interior of the stability domain are removed via a prescribed damping procedure by construction frkc schemes meet all linear order conditions for nonlinear problems at orders above 2 complex splitting or butcher series composition methods are required linear order conditions of the frkc stability polynomials are verified at orders 2 4 and 6 in numerical experiments comparative studies with existing methods show the second order unsplit frkc2 scheme and higher order 4 and 6 split frkcs schemes are efficient for large moderately stiff problems

an estimate for the curvature of bounded submanifolds 1 statement of the result all manifolds considered in this paper shall be connected of class c smooth and dimension at least 2 im mersions will also be smooth and of codimension at least 1 if m and m are riemannian manifolds and p m m is an isometric immersion with the property that sp m lies in a ball we intend to estimate the supremum of the sectional curvature k of m in terms of the sectional curvature k of m and the radius of the ball this is our result theorem a let ml be a complete riemannian manifold whose scalar curvature is bounded below let m q be a riemannian manifold with q c n 1 and bx a closed normal ball in mt4 q of radius x sup pose sp m j m q is an isometric immersion with the property that

pirock a swiss knife partitioned implicit explicit orthogonal runge kutta chebyshev integrator for stiff diffusion advection reaction problems with or without noise a partitioned implicit explicit orthogonal runge kutta method pirock is proposed for the time integration of diffusion advection reaction problems with possibly severely stiff reaction terms and stiff stochastic terms the diffusion terms are solved by the explicit second order orthogonal chebyshev method rock2 while the stiff reaction terms solved implicitly and the advection and noise terms solved explicitly are integrated in the algorithm as finishing procedures it is shown that the various coupling between diffusion reaction advection and noise can be stabilized in the pirock method the method implemented in a single black box code that is fully adaptive provides error estimators for the various terms present in the problem and requires from the user solely the right hand side of the differential equation numerical experiments and comparisons with existing chebyshev methods imex methods and partitioned methods show the efficiency and flexibility of our new algorithm

w methods to stabilize standard explicit runge kutta methods in the time integration of advection diffusion reaction pdes a technique to stabilize standard explicit runge kutta methods by associating them with w methods is proposed the main point to get the associated family of w methods for a given explicit runge kutta method is to require commutativity for the coefficient matrices of the w method in order to reduce the large number of order conditions that must be satisfied to get a pre fixed order based on this idea for any given explicit four stage runge kutta method of order four two uniparametric families of third order w methods are obtained the free parameter can be used to increase the stability regions of the w methods in case of d 1 splittings in the derivative function when a von neumann stability analysis is carried out additionally it is possible to find l stable row methods w methods with exact jacobian for some specific values of the free parameter the new family of w methods is also equipped with the splitting provided by the approximate matrix factorization amf which converts a w method into some kind of adi method alternating direction implicit method the amf w methods so obtained are mainly used to solve large time dependent pde systems in 2d or 3d spatial variables discretized in space by using finite differences or finite volumes some stability properties of the family of amf w methods are also supplied for the case of d splittings d 1 numerical experiments in connection with the proposed amf w methods on a few interesting stiff problems coming from pde discretizations illustrate the stabilization approach in comparison with some relevant methods in the literature

internal error propagation in explicit runge kutta methods in practical computation with runge kutta methods the stage equations are not satisfied exactly due to roundoff errors algebraic solver errors and so forth we show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well known methods we perform a general analysis of internal error propagation emphasizing that it depends significantly on how the method is implemented we show that for a fixed method essentially any set of internal stability polynomials can be obtained by modifying the implementation details we provide bounds on the internal error amplification constants for some classes of methods with many stages including strong stability preserving methods and extrapolation methods these results are used to prove error bounds in the presence of roundoff or other internal errors

convergence properties of the runge kutta chebyshev method the runge kutta chebyshev method is ans stage runge kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations method of lines the method possesses an extended real stability interval with a length proportional tos 2 the method can be applied withs arbitrarily large which is an attractive feature due to the proportionality of withs 2 the involved stability property here is internal stability internal stability has to do with the propagation of errors over the stages within one single integration step this internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space time grid refinement for a model class of linear problems we prove convergence under the sole condition that the necessary time step restriction for stability is satisfied these error bounds are valid for anys and independent of the stiffness of the problem numerical examples are given to illustrate the theoretical results

an implicit explicit runge kutta chebyshev scheme for diffusion reaction equations an implicit explicit imex extension of the explicit runge kutta chebyshev rkc scheme designed for parabolic pdes is proposed for diffusion reaction problems with severely stiff reaction terms the imex scheme treats these reaction terms implicitly and diffusion terms explicitly within the setting of linear stability theory the new imex scheme is unconditionally stable for reaction terms having a jacobian matrix with a real spectrum for diffusion terms the stability characteristics remain unchanged a numerical comparison for a stiff nonlinear radiation diffusion problem between an rkc solver an imex rkc solver and the popular implicit bdf solver vodpk using the krylov solver gmres illustrates the excellent performance of the new scheme

pouzet runge kutta chebyshev method for volterra integral equations of the second kind we construct a stabilized runge kutta method of pouzet type pouzet runge kutta chebyshev method prkc for nonlinear stiff volterra integral equations of the second kind the prkc method is an explicit second order scheme which possesses extended stability region due to the first kind chebyshev polynomial used to establish the stability function

global existence and nonexistence for a class of degenerate parabolic systems this paper investigates the finite time blow up and global existence of the positive solution of the nonlinear degenerate parabolic system ut f1 u u a1v vt f2 v v a2u x t 0 with homogeneous dirichlet boundary data a necessary and sufficient condition for global existence of the solution is derived

efficient variable stiffness methods for cooling of hot rolled steel sections abstract an industrial finite element package for the simulation of alternative cooling strategies for hot rolled steel sections has been enhanced by the incorporation of a variable stiffness second order time integration scheme based on a specially developed family of extended stability explicit runge kutta methods and an l stable semi implicit formula the integration scheme uses local error estimation to vary step size and runge kutta method selection is achieved by monitoring the numerical stiffness through the solution period using a computationally inexpensive estimate for the spectral radius of the system jacobian numerical tests on a range of section cooling problems indicate that the variable stiffness rk code developed is considerably more efficient than standard first order integration methods for the levels of accuracy required for industrial simulations the efficiency compares favorably with other high efficiency codes such as the variable order backward differentiation formulae vode and the nearly stiff rkc explicit solver

curvature estimates for submanifolds immersed into horoballs and horocylinders abstract we prove mean and sectional curvature estimates for submanifolds confined into either a horocylinder of n l or a horoball of n where n is a cartan hadamard manifold with pinched curvature thus these submanifolds behave in many respects like submanifolds immersed into compact balls and into cylinders over compact balls the proofs rely on the hessian comparison theorem for the busemann function

unconditional convergence of some crank nicolson lod methods for initial boundary value problems in this paper convergence properties are discussed for some locally one dimensional lod splitting methods applied to linear parabolic initialboundary value problems we shall consider unconditional convergence where both the stepsize in time and the meshwidth in space tend to zero independently of each other

finite volume element approximation of an inhomogeneous brusselator model with cross diffusion this paper is concerned with the study of pattern formation for an inhomogeneous brusselator model with cross diffusion modeling an autocatalytic chemical reaction taking place in a three dimensional domain for the spatial discretization of the problem we develop a novel finite volume element fve method associated to a piecewise linear finite element approximation of the cross diffusion system we study the main properties of the unique equilibrium of the related dynamical system a rigorous linear stability analysis around the spatially homogeneous steady state is provided and we address in detail the formation of turing patterns driven by the cross diffusion effect in addition we focus on the spatial accuracy of the fve method and a series of numerical simulations confirm the expected behavior of the solutions in particular we show that depending on the spatial dimension the magnitude of the cross diffusion influences the selection of spatial patterns

accelerating the convergence of spectral deferred correction methods in the recent paper by dutt greengard and rokhlin a variant of deferred or defect correction methods is presented which couples gaussian quadrature with the picard integral equation formulation of the initial value ordinary differential equation the resulting spectral deferred correction sdc methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy in this paper we show that for linear problems the iterations in the sdc algorithm are equivalent to constructing a preconditioned neumann series expansion for the solution of the standard collocation discretization of the ode this observation is used to accelerate the convergence of sdc using the gmres krylov subspace method for nonlinear problems the gmres acceleration is coupled with a linear implicit approach stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy efficiency and stability of the original sdc approach furthermore preliminary numerical experiments show that accelerating the convergence of sdc methods can effectively eliminate the order reduction previously observed for stiff ode systems

the strong halfspace theorem for minimal surfaces d hoffman and w h meeks iii department of mathematics university of massachusetts amherst ma 01003 usa in 6 jorge and xavier constructed examples of complete nonplanar minimal surfaces contained between two parallel planes in 3 recently rosenberg and toubiana have found complete minimally immersed annuli that are proper in an open slab 1 12 on the basis of these and other results the question has arisen as to whether or not there could exist a properly immersed minimal surface in 3 that was contained between two parallel planes the answer is negative an immediate consequence of the following theorem

convergence and order reduction of runge kutta schemes applied to evolutionary problems in partial differential equations we address the question of convergence of fully discrete runge kutta approximations we prove that under certain conditions the order in time of the fully discrete scheme equals the conventional order of the runge kutta formula being used however these conditions which are necessary for the result to hold are not natural as a result in many problems the order in time will be strictly smaller than the conventional one a phenomenon called order reduction this phenomenon is extensively discussed both analytically and numerically as distinct from earlier contributions we here treat explicit runge kutta schemes although our results are valid for both parabolic and hyperbolic problems the examples we present are therefore taken from the hyperbolic field as it is in this area that explicit discretizations are most appealing

an adi extrapolated crank nicolson orthogonal spline collocation method for nonlinear reaction diffusion systems an alternating direction implicit adi orthogonal spline collocation osc method is described for the approximate solution of a class of nonlinear reaction diffusion systems its efficacy is demonstrated on the solution of well known examples of such systems specifically the brusselator gray scott gierer meinhardt and schnakenberg models and comparisons are made with other numerical techniques considered in the literature the new adi method is based on an extrapolated crank nicolson osc method and is algebraically linear it is efficient requiring at each time level only o n operations where n is the number of unknowns moreover it is shown to produce approximations which are of optimal global accuracy in various norms and to possess superconvergence properties

a finite element semi lagrangian explicit runge kutta chebyshev method for convection dominated reaction diffusion problems explicit runge kutta chebyshev methods have proved to be efficient for reaction diffusion problems of moderate stiffness in this paper we extend such an efficiency to convection dominated reaction diffusion problems by giving a formulation of these methods in a semi lagrangian framework using c0 finite elements of degree m 2 as the space discretization method we also study the convergence in the l2 norm of the methods proposed in this paper

stepsize restrictions for stability of one step methods in the numerical solution of initial value problems this paper deals with the analysis of general one step methods for the numerical solution of initial boundary value problems for stiff ordinary and partial differential equations restrictions on the stepsize are derived that are necessary and sufficient for the rate of error growth in these methods to be of moderate size these restrictions are related to disks contained in the stability region of the method and the errors are measured with arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of a diffusion convection problem where the error growth is measured with the maximum norm

solving parabolic integro differential equations by an explicit integration method vasudeva murthy a s and j g verwer solving parabolic integro differential equations by an explicit integration method journal of computational and applied mathematics 39 1992 121 132 the subject of this note is the numerical integration in time of nonclassical parabolic initial boundary value problems which involve nonlocal integral terms over the spatial domain the integral terms may appear in the boundary conditions and or in the governing partial differential equation itself these terms generally complicate the application of standard methods our purpose here is to draw attention to an explicit method the runge kutta chebyshev method whose application remains straightforward in many cases of practical interest we discuss two numerical examples to illustrate this a comparison is made respectively with the implicit bdf code dassl and with the keller box scheme

the mean curvature of cylindrically bounded submanifolds we give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder b r times mathbb r ell in a product riemannian manifold n n ell times mathbb r ell it follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in euclidean space cannot be proper if the circular base is of sufficiently small radius in particular any possible counterexample to a conjecture of calabi on complete minimal hypersurfaces cannot be proper as another application of our method we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature

on the mean exit time from a minimal submanifold soit m m une sous variete minimale immergee d une variete de riemann n n et on considere le mouvement brownien sur une boule reguliere r cm de rayon exterieur r le temps de sortie moyen pour le mouvement a partir d un point xe r est appele e r x on trouve des fonctions support pointues pour e r x dans plusieurs cas distincts

convergence analysis of one step schemes in the method of lines we present an expository account of some fundamental results concerning the analysis of one step schemes for semidiscretizations of evolutionary problems in partial differential equations in the paper the emphasis lies in the interplay between the stability and convergence properties of the fully discrete scheme and those of the ordinary differential equations solver much attention is paid to the phenomenon of order reduction

on a degenerate heat equation with a singular potential using a new method we generalize the blow up and existence result from p baras and j a goldstein 1984 trans amer math soc 284 121 139 to heat equations on the heisenberg group in doing so we need to overcome the difficulty that the equation in this case is both degenerate and of variable coefficients comparing with the euclidean case an interesting new result is that solutions can blow up even when the singularity of the potential is weaker than the inverse square of the distance function

on stabilized integration for time dependent pdes an integration method is discussed which has been designed to treat parabolic and hyperbolic terms explicitly and stiff reaction terms implicitly the method is a special two step form of the one step imex implicit explicit rkc runge kutta chebyshev method the special two step form is introduced with the aim of getting a non zero imaginary stability boundary which is zero for the one step method having a non zero imaginary stability boundary allows for example the integration of pure advection equations space discretized with centered schemes the integration of damped or viscous wave equations the integration of coupled sound and heat flow equations etc for our class of methods it also simplifies the choice of temporal step sizes satisfying the von neumann stability criterion by embedding a thin long rectangle inside the stability region embedding rectangles or other tractable domains with this purpose is an idea of wesseling

runge kutta chebyshev projection method in this paper a fully explicit stabilized projection method called the runge kutta chebyshev rkc projection method is presented for the solution of incompressible navier stokes systems this method preserves the extended stability property of the rkc method for solving odes and it requires only one projection per step an additional projection on the time derivative of the velocity is performed whenever a second order approximation for the pressure is desired we demonstrate both by numerical experiments and by order analysis that the method is second order accurate in time for both the velocity and the pressure being explicit the rkc projection method is easy to implement and to parallelize hence it is an attractive candidate for the solution of large scale moderately stiff diffusion like problems

efficient integration of strangeness free non stiff differential algebraic equations by half explicit methods numerical integration methods for nonlinear differential algebraic equations daes in strangeness free form are studied in particular half explicit methods based on popular explicit methods like one leg methods linear multistep methods and runge kutta methods are proposed and analyzed compared with well known implicit methods for daes these half explicit methods demonstrate their efficiency particularly for a special class of semi linear matrix valued daes which arise in the numerical computation of spectral intervals for daes numerical experiments illustrate the theoretical results

some nonlinear equations of the nonstable filtration type and embedding theorems abstract in this article we consider some mixed problems with implicit degenerate equations for their investigation we study certain properties of a class of pn spaces connected with nonlinear boundary value problems further using the results obtained we prove the solvability of some nonlinear boundary value problems with various elliptic parabolic equations and also a boundary problem for an equation of the nonstable filtration type

weak solutions of the porous medium equation we show that if u 0 u l loc m r n 1 solves u t u m m 1 in the sense of distributions then u is locally holder continuous in

an estimate for the sectional curvature of cylindricalli bounded submanifolds we give sharp sectional curvature estimates for complete immersed cylindrically bounded m submanifolds m n rl n l 2m 1 provided that either is proper with the second fundamental form with certain controlled growth or m has scalar curvature with strong quadratic decay this latter gives a non trivial extension of the jorge koutrofiotis theorem 8 mathematics subject classification 2000 53c42

vanishing solutions of anisotropic parabolic equations with variable nonlinearity abstract we study the property of finite time vanishing of solutions of the homogeneous dirichlet problem for the anisotropic parabolic equations u t i 1 n d i a i x t u d i u p i x t 2 d i u c x t u x t 2 u f x t with variable exponents of nonlinearity p i x t x t 1 we show that the solutions of this problem may vanish in a finite time even if the equation combines the directions of slow and fast diffusion and estimate the extinction moment in terms of the data if the solution does not identically vanish in a finite time we estimate the rate of vanishing of the solution as t we establish conditions on the nonlinearity exponents which guarantee vanishing of the solution at a finite instant even if the equation eventually transforms into the linear one

rkc an explicit solver for parabolic pdes the fortran program rkc is intended for the time integration of parabolic partial differential equations discretized by the method of lines it is based on a family of runge kutta chebyshev formulas with a stability bound that is quadratic in the number of stages remarkable properties of the family make it possible for the program to select at each step the most efficient stable formula as well as the most efficient step size moreover they make it possible to evaluate the explicit formulas in just a few vectors of storage these characteristics of the program make it especially attractive for problems in several spatial variables rkc is compared to the bdf solver vodpk on two test problems in three spatial variables

an essentially non oscillatory semi lagrangian method for tidal flow simulations we develop an essentially non oscillatory semi lagrangian method for solving two dimensional tidal flows the governing equations are derived from the incompressible navier stokes equations with assumptions of shallow water flows including bed frictions eddy viscosity wind shear stresses and coriolis forces the method employs the modified method of characteristics to discretize the convective term in a finite element framework limiters are incorporated in the method to reconstruct an essentially non oscillatory algorithm at minor additional cost the central idea consists in combining linear and quadratic interpolation procedures using nodes of the finite element where departure points are localized the resulting semi discretized system is then solved by an explicit runge kutta chebyshev scheme with extended stages this scheme adds in a natural way a stabilizing stage to the conventional runge kutta method using the chebyshev polynomials the proposed method is verified for the recirculation tidal flow in a channel with forward facing step we also apply the method for simulation of tidal flows in the strait of gibraltar in both test problems the proposed method demonstrates its ability to handle the interaction between water free surface and bed frictions copyright 2009 john wiley sons ltd

a finite elements method to solve the bloch torrey equation applied to diffusion magnetic resonance imaging the complex transverse water proton magnetization subject to diffusion encoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the multiple compartment bloch torrey partial differential equation pde in addition steady state laplace pdes can be formulated to produce the homogenized diffusion tensor that describes the diffusion characteristics of the medium in the long time limit in spatial domains that model biological tissues at the cellular level these two types of pdes have to be completed with permeability conditions on the cellular interfaces to solve these pdes we implemented a finite elements method that allows jumps in the solution at the cell interfaces by using double nodes using a transformation of the bloch torrey pde we reduced oscillations in the searched for solution and simplified the implementation of the boundary conditions the spatial discretization was then coupled to the adaptive explicit runge kutta chebyshev time stepping method our proposed method is second order accurate in space and second order accurate in time we implemented this method on the fenics c platform and show time and spatial convergence results finally this method is applied to study some relevant questions in diffusion mri