high resolution schemes for conservation laws with locally varying time steps we develop upwind methods which use limited high resolution corrections in the spatial discretization and local time stepping for forward euler and second order time discretizations l infty stability is proven for both time stepping schemes for problems in one space dimension these methods are restricted by a local cfl condition rather than the traditional global cfl condition allowing local time refinement to be coupled with local spatial refinement numerical evidence demonstrates the stability and accuracy of the methods for problems in both one and two space dimensions

on the symbolic reduction of processes with cryptographic functions we study the reachability problem for cryptographic protocols represented as processes relying on perfect cryptographic functions we introduce a symbolic reduction system that can handle hashing functions symmetric keys and public keys desirable properties such as secrecy or authenticity are specified by inserting logical assertions in the processes we show that the symbolic reduction system provides a flexible decision procedure for finite processes and a reference for sound implementations the symbolic reduction system can be regarded as a variant of syntactic unification which is compatible with certain set membership constraints for a significant fragment of our formalism we argue that a dag implementation of the symbolic reduction system leads to an algorithm running in nptime thus matching the lower bound of the problem in the case of iterated or finite control processes we show that the problem is undecidable in general and in ptime for a subclass of iterated processes that do not rely on pairing our technique is based on rational transductions of regular languages and it applies to a class of processes containing the ping pong protocols presented in dolev et al inform comput 55 1982 57

on conservation at grid interfaces this paper considers the solution of hyperbolic systems of conservation laws on discontinuous grids in particular we consider what happens to conservation at grid interfaces a procedure is presented to derive conservative difference approximations at the grid interfaces for two dimensional grids which overlap in an arbitrary configuration the same procedures are applied to compute interface formulas for grids which are refined in space and or time and for continuous grids where a switch in the scheme causes the discontinuity

a method to discern complexity in two dimensional patterns generated by coupled map lattices complex patterns generated by the time evolution of a one dimensional digitalized coupled map lattice are quantitatively analyzed a method for discerning complexity among different patterns is implemented the quantitative results indicate two zones in parameter space where the dynamics show the most complex patterns these zones are located on the two edges of an absorbent region where the system displays spatio temporal intermittency

proof techniques for cryptographic processes contextual equivalences for cryptographic process calculi like the spi calculus can be used to reason about correctness of protocols but their definition suffers from quantification over all possible contexts here we focus on two such equivalences namely may testing and barbed equivalence and investigate tractable proof methods for them to this aim we design an enriched labelled transition system where transitions are constrained by the knowledge the environment has of names and keys the new transition system is then used to define a trace equivalence and a weak bisimulation equivalence that avoid quantification over contexts our main results are soundness and completeness of trace and weak bisimulation equivalence with respect to may testing and barbed equivalence respectively they lead to more direct proof methods for equivalence checking the use of these methods is illustrated with a few examples concerning implementation of secure channels and verification of protocol correctness

on the existence and decidability of unique decompositions of processes in the applied π calculus unique decomposition has been a subject of interest in process algebra for a long time for example in bpp 1 or ccs 2 3 as it provides a normal form and useful cancellation properties we provide two parallel decomposition results for subsets of the applied calculus we show that every closed normed i e with a finite shortest complete trace process p can be decomposed uniquely into prime factors p i with respect to strong labeled bisimilarity i e such that p l p 1 p n moreover we prove that closed finite processes can be decomposed uniquely with respect to weak labeled bisimilarity we also investigate whether efficient algorithms that compute the unique decompositions exist the simpler problem of deciding whether a process is in its unique decomposition form is undecidable in general in both cases due to potentially undecidable equational theories moreover we show that the unique decomposition remains undecidable even given an equational theory with a decidable word problem

temporal adaptive euler navier stokes algorithm involving unstructured dynamic meshes a temporal adaptive algorithm for the time integration of the two dimensional euler or navier stokes equations is presented the flow solver involves an upwind flux split spatial discretization for the convective terms and central differencing for the shear stress and heat flux terms on an unstructured mesh of triangles the temporal adaptive algorithm is a time accurate integration procedure that allows flows with high spatial and temporal gradients to be computed efficiently by advancing each grid cell near its maximum allowable time step

space time adaptive wavelet methods for parabolic evolution problems with respect to space time tensor product wavelet bases parabolic initial boundary value problems are equivalently formulated as bi infinite matrix problems adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate in case the spatial domain is of product type the use of spatial tensor product wavelet bases is proved to overcome the so called curse of dimensionality i e the reduction of the convergence rate with increasing spatial dimension

a posteriori analysis of a multirate numerical method for ordinary differential equations in this paper we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model the main purpose of this paper is to present a hybrid a priori a posteriori error analysis that accounts for the effects of projections between the coarse and fine scale discretizations the use of only a finite number of iterations in the iterative solution of the discrete equations the numerical error arising in the solution of each component and the effects on stability arising from the multirate solution the hybrid estimate has the form of a computable a posteriori leading order expression and a provably higher order a priori expression we support this estimate by an a priori convergence analysis we present several examples illustrating the accuracy of multirate integration schemes and the accuracy of the a posteriori estimate

a fisher gradient complexity in systems with spatio temporal dynamics abstract we define a benchmark for definitions of complexity in systems with spatio temporal dynamics and employ it in the study of collective motion we show that lmc s complexity displays interesting properties in such systems while a statistical complexity model scm based on autocorrelation reasonably meets our perception of complexity however this scm is not as general as desirable as it does not merely depend on the system s probability distribution function inspired by the notion of fisher information we develop a scm candidate which we call the fisher gradient complexity which exhibits nice properties from the viewpoint of our benchmark

the fast adaptive composite grid fac method for elliptic equation the fast adaptive composite grid fag method is a systematic process for solving differential boundary value problems fac uses global and local uniform grids both to define the composite grid problem and to interact for its fast solution it can with little added cost substantially improve accuracy of the coarse grid solution and is very suitable for vector and parallel computation this paper develops both the theoretical and practical aspects of fac as it applies to elliptic problems 1 introduction the need for local resolution in physical models occurs frequently in practice special local features of the forcing function operator coefficients boundary and boundary conditions can demand resolution in restricted regions of the domain that is much finer than the required global resolution it is important that the discretization and solution processes account for this locally that is that the local phenomena do not precipitate a dramatic increase in the overall computation unfortunately this objective of efficiently adapting to local features is often in conflict with the solution process equation solvers can degrade or even fail to apply in the presence of varying discretization scales data structures that account for irregular grids can be cumbersome the computer architecture may not be able to effectively account for grid irregularity e g vectorizability may be inhibited etc in fact even the discretization process itself may find difficulty with this objective for finite differences it is problematic to develop accurate difference formulae for irregular grids for finite elements this objective is reflected in the substantial overhead costs needed to automate the discretization the fast adaptive composite grid method fac 11 is a discretization and solution method designed to achieve efficient local resolution by constructing the

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

limitations of adaptive mesh refinement techniques for singularly perturbed problems with a moving interior layer in a composed domain on an axis r with the moving interface boundary between two subdomains we consider an initial value problem for a singularly perturbed parabolic reaction diffusion equation in the presence of a concentrated source on the interface boundary monotone classical difference schemes for problems from this class converge only when e n 1 n0 1 where e is the perturbation parameter n and n0 define the number of mesh points with respect to x on segments of unit length and t therefore in the case of such problems with moving interior layers it is necessary to develop special numerical methods whose errors depend rather weakly on the parameter e and in particular are independent of e i e e uniformly convergent methods in this paper we study schemes on adaptive meshes which are locally condensing in a neighbourhood of the set that is the trajectory of the moving source it turns out that in the class of difference schemes consisting of a standard finite difference operator on rectangular meshes which are a priori or a posteriori locally condensing in x and t there are no schemes that converge e uniformly and in particular even under the condition e n 2 n0 2 if the total number of the mesh points between the cross sections x0 and x0 1 for any x0 r has order of nn0 thus the adaptive mesh refinement techniques used directly do not allow us to widen essentially the convergence range of classical numerical methods on the other hand the use of condensing meshes but in a local coordinate system fitted to the set makes it possible to construct schemes which converge e uniformly for n n0 such a scheme converges at the rate o n 1 ln n n0 1

numerical experiments with the osher upwind scheme for the euler equations the osher algorithm for solving the euler equations is an upwind finite difference procedure that is derived by combining the salient features of the theory of conservation laws and the mathematical theory of characteristi cs for hyperbolic systems of equations a first order accurate version of the numerical method was derived by osher circa 1980 for the one dimensional non isentropic euler equations in cartesian coordinates in this paper the extension of the scheme to arbitrary two dimensional geometries is explained results are then presented for several example problems in one and two dimensions future work will include extension of the method to second order accuracy and the development of implicit time differencing for the osher algorithm

an error estimate for finite volume methods for multidimensional conservation laws in this paper an l ll error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained these methods can be formally high order accurate and are defined on general triangulations the error is proven to be of order ft 4 where h represents the size of the mesh via an extension of kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con tinuity in time are needed the result is new even for the finite volume method constructed from monotone numerical flux functions

testing equivalences for processes abstract given a set of processes and a set of tests on these processes we show how to define in a natural way three different equivalences on processes these equivalences are applied to a particular language ccs we give associated complete proof systems and fully abstract models these models have a simple representation in terms of trees

convergence of the finite volume method for multidimensional conservation laws we establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux functions our technique applies with a fairly unrestrictive assumption on the triangulations flat elements are allowed and to lipschitz continuous flux functions we treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of kruzkov the proof of convergence is based on a convergence framework coquel and lefloch math comp 57 1991 pp 169 210 and j numer anal 30 1993 pp 675 700 from a convex decomposition of the scheme we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities these estimates are shown to be sufficient for the passage to the limit in the discrete equation convergence follows from diperna s uniqueness result in the class of entropy measure valued solutions

deciding equivalence based properties using constraint solving formal methods have proved their usefulness for analyzing the security of protocols most existing results focus on trace properties like secrecy or authentication there are however several security properties which cannot be defined or cannot be naturally defined as trace properties and require a notion of behavioural equivalence typical examples are anonymity privacy related properties or statements closer to security properties used in cryptography in this paper we consider three notions of equivalence defined in the applied pi calculus observational equivalence may testing equivalence and trace equivalence first we study the relationship between these three notions we show that for determinate processes observational equivalence actually coincides with trace equivalence a notion simpler to reason with we exhibit a large class of determinate processes called simple processes that capture most existing protocols and cryptographic primitives while trace equivalence and may testing equivalence seem very similar we show that may testing equivalence is actually strictly stronger than trace equivalence we prove that the two notions coincide for image finite processes such as processes without replication second we reduce the decidability of trace equivalence for finite processes to deciding symbolic equivalence between sets of constraint systems for simple processes without replication and with trivial else branches it turns out that it is actually sucient to decide symbolic equivalence between pairs of positive constraint systems thanks to this reduction and relying on a result first proved by m baudet this yields the first decidability result of observational equivalence for a general class of equational theories for processes without else branch nor replication moreover based on another decidability result for deciding equivalence between sets of constraint systems we get decidability of trace equivalence for processes with else branch for standard primitives

a moving mesh numerical method for hyperbolic conservation laws we show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in l1 r to within o n 2 by a piecewise linear function with o fn nodes the nodes are moved according to the method of characteristics we also show that a previous method of dafermos which uses piecewise constant approxima tions is accurate to o n 1 these numerical methods for conservation laws are the first to have proven convergence rates of greater than o fn 1 2 1 introduction it is well known that the solution of the hyperbolic conservation law

an efficient local time stepping scheme for solution of nonlinear conservation laws we develop an efficient local time stepping algorithm for the method of lines approach to numerical solution of transient partial differential equations the need for local time stepping arises when adaptive mesh refinement results in a mesh containing cells of greatly different sizes the global cfl number and hence the global time step are defined by the smallest cell size this can be inefficient as a few small cells may impose a restrictive time step on the whole mesh a local time stepping scheme allows us to use the local cfl number which reduces the total number of function evaluations the algorithm is based on a second order runge kutta time integration its important features are a small stencil and the second order accuracy in the l 2 and l norms

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

finite point solver for the simulation of 2 d laminar incompressible unsteady flows abstract a finite point solver has been developed to simulate 2 d laminar incompressible unsteady flow fields in the low reynolds number regime reynolds of the order of 100 1000 grid points without connectivity in the classic sense are scattered all over the computational domain and a certain flow functional behaviour is assumed to hold in a small region around each point two types of functional behaviour have been tested a truncated and a full second order taylor expansion the expansion coefficients are obtained by using a least squares approximation in the cloud of points located around the point under consideration an artificial compressibility formulation has been used for the continuity equation the validity of this approach to deal with unsteady flow situations has also been discussed time integration has been performed by using a lax wendroff formulation because of numerical stability concerns three artificial viscosity terms have been introduced concerning solver validation a detailed sensitivity analysis is performed with regard to the three artificial viscosity terms the pseudo compressibility coefficient the number of grid points in the flow field and the reynolds number the results obtained strouhal number average drag and lift root mean square values were compared with experimental results and also with the numerical results obtained by other researchers application of the solver to the steady regime is also presented

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

grid adaptive algorithms for the solution of the euler equations on irregular grids abstract the effect of grid irregularity on the accuracy of algorithms for one dimensional unsteady flow is investigated it is shown how standard second order accurate algorithms for equally spaced grids may be extended to act as first order accurate algorithms on an irregular grid the loss of accuracy on irregular grids is found to cause a significant reduction in the quality of the results to obtain solutions comparable in quality with second order accurate algorithms on equally spaced grids a class of second order accurate algorithms is derived which are conservative and compatible on randomly spaced grids they are shown to give solutions to the equations similar in quality to those obtainable on equally spaced grids

solving the advection diffusion equation on unstructured meshes with discontinuous mixed finite elements and a local time stepping procedure explicit schemes are known to provide less numerical diffusion in solving the advection diffusion equation especially for advection dominated problems traditional explicit schemes use fixed time steps restricted by the global cfl condition in order to guarantee stability this is known to slow down the computation especially for heterogeneous domains and or unstructured meshes to avoid this problem local time stepping procedures where the time step is allowed to vary spatially in order to satisfy a local cfl condition have been developed in this paper a local time stepping approach is used with a numerical model based on discontinuous galerkin mixed finite element methods to solve the advection diffusion equation the developments are detailed for general unstructured triangular meshes numerical experiments are performed to show the efficiency of the numerical model for the simulation of i the transport of a solute on highly unstructured meshes and ii density driven flow where the velocity field changes at each time step the model gives stable results with significant reduction of the computational cost especially for the non linear problem moreover numerical diffusion is also reduced for highly advective problems copyright 2009 john wiley sons ltd

an iterative method for elliptic problems on regions partitioned into substructures some new preconditioners for discretizations of elliptic boundary problems are studied with these preconditioners the domain under consideration is broken into subdomains and preconditioners are defined which only require the solution of matrix problems on the subdomains analytic estimates are given which guarantee that under appropriate hypotheses the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns numerical examples are presented which illustrate the theoretically predicted iterative convergence rates

the moving grid method for nonlinear hyperbolic conservation laws numerically moving grids are often found to improve the resolution of approximate solutions to nonlinear hyperbolic conservation laws in this paper we show how to modify many standard numerical flux functions to incorporate this idea a generalized godunov type scheme is shown in the scalar case to converge to the entropy satisfying solution as the mesh size tends to zero a courant number of one half is found necessary in general and proved to be sufficient to obtain convergence convergence for methods based upon any two point conservative and monotone numerical flux function is also shown in the scalar case

parallel domain decomposition procedures of improved d d type for parabolic problems two parallel domain decomposition procedures for solving initial boundary value problems of parabolic partial differential equations are proposed one is the extended d d type algorithm which extends the explicit implicit conservative galerkin domain decomposition procedures given in 5 from a rectangle domain and its decomposition that consisted of a stripe of sub rectangles into a general domain and its general decomposition with a net like structure an almost optimal error estimate without the factor h 1 2 given in dawson dupont s error estimate is proved another is the parallel domain decomposition algorithm of improved d d type in which an additional term is introduced to produce an approximation of an optimal error accuracy in l 2 norm

simple time and space adaptation in one dimensional evolutionary partial differential equations a simple technique for time and space adaptation in one dimensional evolutionary partial differential equations suggested by sanz serna and christie 1 is tested it is found that the equidistribution of hu xi greatly improves on the equidistribution of arc length used by those authors the time step control is found to perform poorly in the integration of rough solutions and the reasons for this behaviour are analysed

local a posteriori error estimates for time dependent hamilton jacobi equations university of minnesota ph d dissertation may 2010 major mathematics advisor prof bernardo cockburn 1 computer file pdf viii 67 pages ill some col

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

deciding knowledge in security protocols under equational theories the analysis of security protocols requires precise formulations of the knowledge of protocol participants and attackers in formal approaches this knowledge is often treated in terms of message deducibility and indistinguishability relations in this paper we study the decidability of these two relations the messages in question may employ functions encryption decryption etc axiomatized in an equational theory one of our main positive results says that deducibility and indistinguishability are both decidable in polynomial time for a large class of equational theories this class of equational theories is defined syntactically and includes for example theories for encryption decryption and digital signatures we also establish general decidability theorems for an even larger class of theories these theorems require only loose abstract conditions and apply to many other useful theories for example with blind digital signatures homomorphic encryption xor and other associative commutative functions

adaptive techniques for time dependent problems abstract both a priori and a posteriori error estimation techniques have played an important role in the adaptivity of numerical methods for static problems with important localized phenomena timedependent problems with localized temporal effects offer an even greater challenge to effective adaptivity the need for flexible data structures and solution algorithms to accommodate moving patch refinement to follow moving sharp fronts in fluid flow applications is discussed in parabolic problems often transient temporal effects are generated by spatially fixed but dynamic source terms local time stepping methods are presented to treat these transient phenomena stability and accuracy of the methods are considered results of numerical experiments are presented that indicate the relative roles of the local temporal and spatial discretizations in the performance of the methods

runge kutta based explicit local time stepping methods for wave propagation locally refined meshes severely impede the efficiency of explicit runge kutta rk methods for the simulation of time dependent wave phenomena by taking smaller time steps precisely where the smallest elements are located local time stepping lts methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh starting from classical or low storage explicit rk methods explicit lts methods of arbitrarily high accuracy are derived when combined with an essentially diagonal finite element mass matrix the resulting time marching schemes retain the high accuracy stability and efficiency of the original rk methods while circumventing the geometry induced stiffness numerical experiments with continuous and discontinuous galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these lts rk methods

subspace correction methods for nonsymmetric parabolic problems abstract a convergence proof for general subspace correction techniques applied to an abstract nonsymmetric parabolic equation is given one of the main concerns is to give a unified convergence analysis for domain decomposition and multigrid methods for parabolic problems the analysis is also valid for nonsymmetric problems for second order parabolic problems the convection can dominate the diffusion the algorithms can be applied to domain decomposition methods with or without the coarse mesh for applications to multigrid methods the coarsest mesh does not need to be very coarse a relation between the coarse mesh size and the time step is needed to get a convergence rate independent of the mesh the number of iterations at each time step for the algorithms is also estimated some numerical experiments are presented for domain decomposition methods with minimum overlap which support the theoretical predictions the algorithms are able to capture the sharp traveling shocks for convection dominated problems

determinancy observation equivalence trace equivalence abstract if an experiment s is conducted on a parallel process p then in general different processes may result from the experiment due to the nondeterministic behaviour of p in the notation of milner 1980 p p for different p process p is called determinate if the resulting processes are all equivalent i e if p p and p p then p and p are equivalent this means that although p behaves nondeterministically this cannot be detected by an observer of p let denote observation equivalence used in ccs milner 1980 let f denote the much weaker failure equivalence used for csp hoare et al 1981 brookes 1983 and let t denote the still weaker trace equivalence we show that the three corresponding notions of determinancy are the same and that for determinate processes f and t are the same determinacy is preserved under and f but not under t

space time adaptive wavelet methods for optimal control problems constrained by parabolic evolution equations an adaptive algorithm based on wavelets is proposed for the efficient numerical solution of a control problem governed by a linear parabolic evolution equation first the constraints are represented by means of a full weak space time formulation as a linear system in ell 2 in wavelet coordinates following a recent approach by schwab and stevenson second a quadratic cost functional involving a tracking type term for the state and a regularization term for the distributed control is also formulated in terms of ell 2 sequence norms of wavelet coordinates this functional serves as a representer for a functional involving different sobolev norms with possibly nonintegral smoothness parameter standard techniques from optimization are then used to derive the first order necessary conditions as a coupled system in ell 2 coordinates for this purpose an adaptive method is proposed which can be interpreted as an inexact gradient method for the control in each iteration step the primal and adjoint systems are solved up to a prescribed accuracy by the adaptive algorithm it is shown that the adaptive algorithm converges moreover the algorithm is proved to be asymptotically optimal the convergence rate achieved for computing each of the components of the solution state adjoint state and control up to a desired target tolerance is asymptotically the same as the wavelet best n term approximation of each solution component and the total computational work is proportional to the number of computational unknowns

hierarchical combination of intruder theories recently automated deduction tools have proved to be very effective for detecting attacks on cryptographic protocols these analysis can be improved for finding more subtle weaknesses by a more accurate modelling of operators employed by protocols several works have shown how to handle a single algebraic operator associated with a fixed intruder theory or how to combine several operators satisfying disjoint theories however several interesting equational theories such as exponentiation with an abelian group law for exponents remain out of the scope of these techniques this has motivated us to introduce a new notion of hierarchical combination for non disjoint intruder theories and to show decidability results for the deduction problem in these theories we have also shown that under natural hypotheses hierarchical intruder constraints can be decided this result applies to an exponentiation theory that appears to be more general than the one considered before

domain decomposition type iterative techniques for parabolic problems on locally refined grids based on an extension of the discontinuous galerkin finite element method discretization schemes for solving parabolic problems on grids with local refinement both in space and in time are constructed the stability of schemes constructed in this way is automatically ensured by the method the construction of two level preconditioners utilizing local timestepping and a global coarse grid solver both on standard rectangular and uniform grids is the main objective of the paper the optimal convergence properties of such two level preconditioners are studied the theory is illustrated by a set of numerical examples

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

interpolation finite difference schemes on grids locally refined in time numerical methods in which the mesh is locally refined are widely used for problems with singularities in the solution in this case approaches with refining the grid in both space and time are being developed in this paper we consider a class of finite difference schemes with local refinement of the grid in time to solve the problems numerically here we compute the numerical solution on a finer time grid in a part of the domain we consider a model dirichlet problem for a second order parabolic equation on a rectangle we analyze the accuracy of completely implicit schemes with the simplest interpolated interface conditions on the boundary of the adaptation domain on the basis of the maximum principle the unconditional convergence of these schemes in the uniform norm is shown and the rate of convergence is analyzed

a lax wendroff type theorem for unstructured quasi uniform grids a well known theorem of lax and wendroff states that if the sequence of approximate solutions to a system of hyperbolic conservation laws generated by a conservative consistent numerical scheme converges boundedly a e as the mesh parameter goes to zero then the limit is a weak solution of the system moreover if the scheme satisfies a discrete entropy inequality as well the limit is an entropy solution the original theorem applies to uniform cartesian grids this article presents a generalization for quasi uniform grids with lipschitz boundary cells uniformly continuous inhomogeneous numerical fluxes and nonlinear inhomogeneous sources the added generality allows a discussion of novel applications like local time stepping grids with moving vertices and conservative remapping a counterexample demonstrates that the theorem is not valid for arbitrary non quasi uniform grids

numerical approximations to nonlinear conservation laws with locally varying time and space grids an explicit time differencing technique is introduced to approximate nonlinear conservation laws this differencing technique links together an arbitrary number of space regimes containing fine and coarse time increments previous stability requirements i e the cfl condition placed a global bound on the size of the time increments for scalar monotone approximations in one space dimension using this variable step time differencing convergence to the correct physical solution is proven given only a local cfl condition

on convergence of a domain decomposition method for a scalar conservation law in this paper we prove convergence of a domain decomposition method for one dimensional scalar conservation laws by dealing carefully with nonconservative terms at the interface of subdomains the method consists of an explicit scheme in some subdomains and an implicit scheme in other subdomains with a numerical flux being the same as the one used in the explicit scheme although such a multidomain algorithm is not strictly conservative the conservation error ce 0 n delta t is equal to mathcal o delta t regardless of the smoothness of the solution finally two test examples are given to validate convergence and the computational efficiency of the present method

an error analysis of conservative space time mesh refinement methods for the one dimensional wave equation we study two space time mesh refinement methods as the one introduced in f collino t fouquet and p joly numer math 95 2003 pp 197 221 the stability of such methods is guaranteed by construction through the conservation of a discrete energy in this paper we show the l2 convergence of these schemes and provide optimal error estimates the proof is based on energy techniques and bootstrap arguments our results are validated with numerical simulations and compared with results from plane wave analysis f collino t fouquet and p joly numer math 95 2003 pp 223 251

a parallel local timestepping runge kutta discontinuous galerkin method with applications to coastal ocean modeling abstract geophysical flows over complex domains often encompass both coarse and highly resolved regions approximating these flows using shock capturing methods with explicit timestepping gives rise to a courant friedrichs lewy cfl timestep constraint this approach can result in small global timesteps often dictated by flows in small regions vastly increasing computational effort over the whole domain one approach for coping with this problem is to use locally varying timesteps in previous work we formulated a local timestepping lts method within a runge kutta discontinuous galerkin framework and demonstrated the accuracy and efficiency of this method on serial machines for relatively small scale shallow water applications for more realistic models involving large domains and highly complex physics the lts method must be parallelized for multi core parallel computers furthermore additional physics such as strong wind forcing can effect the choice of local timesteps in this paper we describe a parallel lts method parallelized using domain decomposition and mpi we demonstrate the method on tidal flows and hurricane storm surge applications in the coastal regions of the western north atlantic ocean

application of implicit explicit high order runge kutta methods to discontinuous galerkin schemes despite the popularity of high order explicit runge kutta erk methods for integrating semi discrete systems of equations erk methods suffer from severe stability based time step restrictions for very stiff problems we implement a discontinuous galerkin finite element method dgfem along with recently introduced high order implicit explicit runge kutta imex rk schemes to overcome geometry induced stiffness in fluid flow problems the imex algorithms solve the non stiff portions of the domain using explicit methods and isolate and solve the more expensive stiff portions using an l stable stiffly accurate explicit singly diagonally implicit runge kutta method esdirk furthermore we apply adaptive time step controllers based on the embedded temporal error predictors we demonstrate in a number of numerical test problems that imex methods in conjunction with efficient preconditioning become more efficient than explicit methods for systems exhibiting high levels of grid induced stiffness

multirate runge kutta schemes for advection equations explicit time integration methods can be employed to simulate a broad spectrum of physical phenomena the wide range of scales encountered lead to the problem that the fastest cell of the simulation dictates the global time step multirate time integration methods can be employed to alter the time step locally so that slower components take longer and fewer time steps resulting in a moderate to substantial reduction of the computational cost depending on the scenario to simulate s osher r sanders numerical approximations to nonlinear conservation laws with locally varying time and space grids math comput 41 1983 321 336 h tang g warnecke a class of high resolution schemes for hyperbolic conservation laws and convection diffusion equations with varying time and pace grids siam j sci comput 26 4 2005 1415 1431 e constantinescu a sandu multirate timestepping methods for hyperbolic conservation laws siam j sci comput 33 3 2007 239 278 in air pollution modeling the advection part is usually integrated explicitly in time where the time step is constrained by a locally varying courant friedrichs lewy cfl number multirate schemes are a useful tool to decouple different physical regions so that this constraint becomes a local instead of a global restriction therefore it is of major interest to apply multirate schemes to the advection equation we introduce a generic recursive multirate runge kutta scheme that can be easily adapted to an arbitrary number of refinement levels it preserves the linear invariants of the system and is of third order accuracy when applied to certain explicit runge kutta methods as base method

moving mesh method with local time step refinement for blow up problems abstract in this paper the dynamical moving mesh method is merged with the local time stepping technique and the new method is applied for blow up problems it has some benefit in approximating an accurate blow up time by starting at any positive initial time t 0 0 our numerical experiment shows that without applying the local time stepping method wrong blow up time will be obtained if the equation integrates from t 0 0

the finite element method for parabolic equations in this first of two papers computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time independent space meshes the effectiveness of the error estimator is related to conditions on the solution regularity mesh family type and asymptotic range for the mesh size for clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used the results extend straight forwardly to systems of equations and higher order elements in one space dimension while the higher dimensional case requires additional considerations the theory presented here provides the basis for the analysis and adaptive construction of time dependent space meshes which is the subject of the second paper computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems

moving mesh methods with locally varying time steps the time steps associated with moving mesh methods are proportional to the smallest mesh size in space and as a result they are very small at each time level for some practical problems the physical phenomena develop dynamically singular or nearly singular solutions in fairly localized regions and therefore the smallest time step at each time level occurs only in these localized regions in this work we will develop a local time stepping algorithm for the moving mesh methods the principal idea will be demonstrated by investigating the nonlinear hyperbolic conservation laws numerical experiments are carried out to demonstrate the efficiency and robustness of the proposed methods

multiadaptive galerkin methods for odes iii a priori error estimates the multiadaptive continuous discontinuous galerkin methods mcgq and mdgq for the numerical solution of initial value problems for ordinary differential equations are based on piecewise polynomial approximation of degree q on partitions in time with time steps which may vary for different components of the computed solution in this paper we prove general order a priori error estimates for the mcgq and mdgq methods to prove the error estimates we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution the estimates then follow from interpolation estimates together with stability estimates for the discrete dual solution

finite difference scheme for parabolic problems on composite grids with refinement in time and space finite difference schemes for transient convection diffusion problems on grids with local refinement in time and space are constructed and studied the construction utilizes a modified upwind approximation and linear interpolation at the slave nodes the proposed schemes are implicit of backward euler type and unconditionally stable error analysis is presented in the maximum norm and convergence estimates are derived for smooth solutions optimal approximation results for ratios between the spatial and time discretization parameters away from the cfl condition are shown finally numerical examples illustrating the theory are given

statistical complexity measures as telltale of relevant scales in emergent dynamics of spatial systems abstract the definition of complexity through statistical complexity measures scm has recently seen major improvements mostly the effort is concentrated in measures on time series we propose a scm definition for spatial dynamical systems our definition is in line with the trend to combine entropy with measures of structure such as disequilibrium we study the behaviour of our definition against the vectorial noise model of collective motion from a global perspective we show how our scm is minimal at both the microscale and macroscale while it reaches a maximum at the ranges that define the mesoscale in this model from a local perspective the scm is minimum both in highly ordered and disordered areas while it reaches a maximum at the edges between such areas these characteristics suggest this is a good candidate for detecting the mesoscale of arbitrary dynamical systems as well as regions where the complexity is maximal in such systems