analysis of asymptotic preserving dg imex schemes for linear kinetic transport equations in a diffusive scaling in this paper some theoretical aspects will be addressed for the asymptotic preserving discontinuous galerkin implicit explicit dg imex schemes recently proposed in j jang f li j m qiu and t xiong high order asymptotic preserving dg imex schemes for discrete velocity kinetic equations in a diffusive scaling http arxiv org abs 1306 0227 2013 submitted for kinetic transport equations under a diffusive scaling we will focus on the methods that are based on discontinuous galerkin dg spatial discretizations with the p k polynomial space and a first order implicit explicit imex temporal discretization and apply them to two linear models the telegraph equation and the one group transport equation in slab geometry in particular we will establish uniform numerical stability with respect to knudsen number varepsilon using energy methods as well as error estimates for any given varepsilon when varepsilon rightarrow 0 a rigorous asymptotic analysis of the schemes is also obta

numerical simulation of three dimensional nonlinear water waves we present an accurate and efficient numerical model for the simulation of fully nonlinear non breaking three dimensional surface water waves on infinite or finite depth as an extension of the work of craig and sulem 19 the numerical method is based on the reduction of the problem to a lower dimensional hamiltonian system involving surface quantities alone this is accomplished by introducing the dirichlet neumann operator which is described in terms of its taylor series expansion in homogeneous powers of the surface elevation each term in this taylor series can be computed efficiently using the fast fourier transform an important contribution of this paper is the development and implementation of a symplectic implicit scheme for the time integration of the hamiltonian equations of motion as well as detailed numerical tests on the convergence of the dirichlet neumann operator the performance of the model is illustrated by simulating the long time evolution of two dimensional steadily progressing waves as well as the development of three dimensional short crested nonlinear waves both in deep and shallow water

discontinuous galerkin finite element approximation of the cahn hilliard equation with convection the paper is concerned with the construction and convergence analysis of a discontinuous galerkin finite element method for the cahn hilliard equation with convection using discontinuous piecewise polynomials of degree p geq1 and backward euler discretization in time we show that the order parameter c is approximated in the broken rm l infty rm h 1 norm with optimal order cal o h p tau the associated chemical potential w phi c gamma 2 delta c is shown to be approximated with optimal order cal o h p tau in the broken rm l 2 rm h 1 norm here phi c frac 1 4 1 c 2 2 is a quartic free energy function and gamma 0 is an interface parameter numerical results are presented with polynomials of degree p 1 2 3

combined hybridizable discontinuous galerkin hdg and runge kutta discontinuous galerkin rk dg formulations for green naghdi equations on unstructured meshes abstract in this paper we introduce some new high order discrete formulations on general unstructured meshes especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations working with a recent family of optimized asymptotically equivalent equations we benefit from the simplified analytical structure of the linear dispersive operators to conveniently reformulate the models as the classical nonlinear shallow water equations supplemented with several algebraic source terms which globally account for the non hydrostatic effects through the introduction of auxiliary coupling variables high order discrete approximations of the main flow variables are obtained with a rk dg method while the trace of the auxiliary variables are approximated on the mesh skeleton through the resolution of second order linear elliptic sub problems with high order hdg formulations the combined use of hybrid unknowns and local post processing significantly helps to reduce the number of globally coupled unknowns in comparison with previous approaches the proposed formulation is then extended to a more complex family of three parameters enhanced green naghdi equations the resulting numerical models are validated through several benchmarks involving nonlinear waves transformations and propagation over varying topographies showing good convergence properties and very good agreements with several sets of experimental data

a duality for modules over monoidal categories of representations of semisimple hopf algebras for a finite dimensional semisimple cosemisimple hopf algebra a and its dual hopf algebra b we set up a natural one to one correspondence between categories with actions of the monoidal categories of representations of a and of b this gives a categorical interpretation of the duality for actions of hopf algebras on algebras

a high order spectral method for nonlinear water waves over moving bottom topography we present a numerical method for simulations of nonlinear surface water waves over variable bathymetry it is applicable to either two or three dimensional flows as well as to either static or moving bottom topography the method is based on the reduction of the problem to a lower dimensional hamiltonian system involving boundary quantities alone a key component of this formulation is the dirichlet neumann operator which in light of its joint analyticity properties with respect to surface and bottom deformations is computed using its taylor series representation we present new stabilized forms for the taylor terms each of which is efficiently computed by a pseudospectral method using the fast fourier transform physically relevant applications are displayed to illustrate the performance of the method comparisons with analytical solutions and laboratory experiments are provided

a modified central discontinuous galerkin method with positivity preserving and well balanced properties for the one dimensional nonlinear shallow water equations abstract in this paper we present a modified central discontinuous galerkin method for solving the nonlinear shallow water equations over variable bottom topography in one dimensional space the proposed method balances exactly the flux gradient and source term in the still water stationary case by adding a correction term to the base scheme and ensures non negativity of the water depth by using special approximations to the bottom together with a positivity preserving limiter meanwhile reduces the computational cost considerably compared with the standard central discontinuous galerkin method by using l 2 projection to obtain an approximation solution numerical tests are presented to illustrate the accuracy and validity of the proposed schemes

stability analysis and a priori error estimates of the third order explicit runge kutta discontinuous galerkin method for scalar conservation laws in this paper we present an analysis of the runge kutta discontinuous galerkin method for solving scalar conservation laws where the time discretization is the third order explicit total variation diminishing runge kutta method we use an energy technique to prove the mathrm l 2 norm stability for scalar linear conservation laws and to obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws quasi optimal order is obtained for general numerical fluxes and optimal order is given for upwind fluxes the theoretical results are obtained for piecewise polynomials with any degree k geq1 under the standard temporal spatial cfl condition tau leq gamma h where h and tau are the element length and time step respectively and the positive constant gamma is independent of h and tau

arbitrary order exactly divergence free central discontinuous galerkin methods for ideal mhd equations ideal magnetohydrodynamic mhd equations consist of a set of nonlinear hyperbolic conservation laws with a divergence free constraint on the magnetic field neglecting this constraint in the design of computational methods may lead to numerical instability or nonphysical features in solutions in our recent work f li l xu s yakovlev central discontinuous galerkin methods for ideal mhd equations with the exactly divergence free magnetic field journal of computational physics 230 2011 4828 4847 second and third order exactly divergence free central discontinuous galerkin methods were proposed for ideal mhd equations in this paper we further develop such methods with higher order accuracy the novelty here is that the well established h div conforming finite element spaces are used in the constrained transport type framework and the magnetic induction equations are extensively explored in order to extract sufficient information to uniquely reconstruct an exactly divergence free magnetic field the overall algorithm is local and it can be of arbitrary order of accuracy numerical examples are presented to demonstrate the performance of the proposed methods especially when they are fourth order accurate

a cache aware algorithm for pdes on hierarchical data structures based on space filling curves competitive numerical algorithms for solving partial differential equations have to work with the most efficient numerical methods like multigrid and adaptive grid refinement and thus with hierarchical data structures unfortunately in most implementations hierarchical data typically stored in trees cause a nonnegligible overhead in data access to overcome this quandary numerical efficiency versus efficient implementation our algorithm uses space filling curves to build up data structures which are processed linearly in fact the only kind of data structure used in our implementation is stacks thus data access becomes very fast even faster than the common access to nonhierarchical data stored in matrices and in particular cache misses are reduced considerably furthermore the implementation of multigrid cycles and or higher order discretizations as well as the parallelization of the whole algorithm become very easy and straightforward on these data structures

high order central discontinuous galerkin finite element methods for the camassa holm equation in this paper we consider the numerical approximation of the camassa holm equation which supports peakon solutions we develop and test a high order central discontinuous galerkin finite element methods for solving this equation in our numerical approach we first reformulate the camassa holm equation into a conservation law coupled with an elliptic equation then we propose a family of high order numerical methods which discretize the conservation law with central discontinuous galerkin methods and the elliptic equation with continuous finite element methods numerical tests are presented to illustrate the accuracy and robustness of the proposed schemes

a note on the convergence of the discontinuous galerkin method for a scalar hyperbolic equation a numerical example is presented which shows that the known l 2 error estimate left u u h right leqq ch k 1 2 left u right k 1 for the discontinuous galerkin method cannot be improved within the class of quasi uniform meshes not even for smooth exact solutions u

a fast and stable well balanced scheme with hydrostatic reconstruction for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom it is a hyperbolic system of conservation laws that approximately describes various geophysical flows such as rivers coastal areas and oceans when completed with a coriolis term or granular flows when completed with friction numerical approximate solutions to this system may be generated using conservative finite volume methods which are known to properly handle shocks and contact discontinuities however in general these schemes are known to be quite inaccurate for near steady states as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions this difficulty can be overcome by using the so called well balanced schemes we describe a general strategy based on a local hydrostatic reconstruction that allows us to derive a well balanced scheme from any given numerical flux for the homogeneous problem whenever the initial solver satisfies some classical stability properties it yields a simple and fast well balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality

conservative discontinuous galerkin methods for the nonlinear serre equations abstract in this paper we develop three conservative discontinuous galerkin dg schemes for the one dimensional nonlinear dispersive serre equations including two conserved schemes for the equations in conservative form and a hamiltonian conserved scheme for the equations in non conservative form one of the schemes owns the well balanced property via constructing a high order approximation to the source term for the serre equations with a non flat bottom topography by virtue of the hamiltonian structure of the serre equations we introduce an hamiltonian invariant and then develop a dg scheme which can preserve the discrete version of such an invariant numerical experiments in different cases are performed to verify the accuracy and capability of these dg schemes for solving the serre equations

a posteriori error estimation for discontinuous galerkin solutions of hyperbolic problems abstract we analyze the spatial discretization errors associated with solutions of one dimensional hyperbolic conservation laws by discontinuous galerkin methods dgms in space we show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a radau polynomial of degree p 1 on each element we also prove that the local and global discretization errors are o x2 p 1 and o x2p 1 at the downwind point of each element this strong superconvergence enables us to show that local and global discretization errors converge as o xp 2 at the remaining roots of radau polynomial of degree p 1 on each element convergence of local and global discretization errors to the radau polynomial of degree p 1 also holds for smooth solutions as p these results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth

upwind residual discretization of enhanced boussinesq equations for wave propagation over complex bathymetries in this paper we consider the solution of the enhanced boussinesq equations of madsen and sorensen 1992 55 by means of residual based discretizations in particular we investigate the applicability of upwind and stabilized variants of continuous galerkin finite element and residual distribution schemes for the simulation of wave propagation and transformation over complex bathymetries these techniques have been successfully applied to the solution of the nonlinear shallow water equations see e g hauke 1998 39 and ricchiuto and bollermann 2009 61 in a first step toward the construction of a hybrid model coupling the enhanced boussinesq equations with the shallow water equations in breaking regions this paper shows that equal order and even low order second upwind stabilized techniques can be used to model non hydrostatic wave propagation over complex bathymetries this result is supported by theoretical truncation and dispersion error analyses and by thorough numerical validation

a pseudo spectral algorithm and test cases for the numerical solution of the two dimensional rotating green naghdi shallow water equations a pseudo spectral algorithm is presented for the solution of the rotating green naghdi shallow water equations in two spatial dimensions the equations are first written in vorticity divergence form in order to exploit the fact that time derivatives then appear implicitly in the divergence equation only a nonlinear equation must then be solved at each time step in order to determine the divergence tendency the nonlinear equation is solved by means of a simultaneous iteration in spectral space to determine each fourier component the key to the rapid convergence of the iteration is the use of a good initial guess for the divergence tendency which is obtained from polynomial extrapolation of the solution obtained at previous time levels the algorithm is therefore best suited to be used with a standard multi step time stepping scheme e g leap frog two test cases are presented to validate the algorithm for initial value problems on a square periodic domain the first test is to verify cnoidal wave speeds in one dimension against analytical results the second test is to ensure that the miles salmon potential vorticity is advected as a parcel wise conserved tracer throughout the nonlinear evolution of a perturbed jet subject to shear instability the algorithm is demonstrated to perform well in each test the resulting numerical model is expected to be of use in identifying paradigmatic behavior in mesoscale flows in the atmosphere and ocean in which both vortical nonlinear and dispersive effects are important

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

an optimal order error estimate for the discontinuous galerkin method in this paper a new approach is developed for analyzing the discontinuous galerkin method for hyperbolic equations for a model problem in r2 the method is shown to converge at a rate 0 hn l when applied with nth degree polynomial approxi mations over a semiuniform triangulation assuming sufficient regularity in the solution

large time existence for 1d green naghdi equations we consider here the 1d green naghdi equations that are commonly used in coastal oceanography to describe the propagation of large amplitude surface waves we show that the solution of the green naghdi equations can be constructed by a standard picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition

a numerical scheme for the green naghdi model abstract in this paper a hybrid numerical method using a godunov type scheme is proposed to solve the green naghdi model describing dispersive shallow water waves the corresponding equations are rewritten in terms of new variables adapted for numerical studies in particular the numerical scheme preserves the dynamics of solitary waves some numerical results are shown and compared to exact and or experimental ones in different and significant configurations a dam break problem and an impact problem where a liquid cylinder is falling to a rigid wall are solved numerically this last configuration is also compared with experiments leading to a good qualitative agreement

a cdg fe method for the two dimensional green naghdi model with the enhanced dispersive property abstract in this work we investigate numerical solutions of the two dimensional shallow water wave using a fully nonlinear green naghdi model with an improved dispersive effect for numerics the green naghdi model is rewritten into a formulation coupling a pseudo conservative system and a set of pseudo elliptic equations since the pseudo conservative system is no longer hyperbolic and its riemann problem can only be approximately solved we consider the utilization of the central discontinuous galerkin method which possesses an important feature of the needlessness of riemann solvers meanwhile the stationary elliptic part will be solved using the finite element method both the well balanced and the positivity preserving features which are highly desirable in the simulation of the shallow water wave will be embedded into the proposed numerical scheme the accuracy and efficiency of the numerical model and method will be illustrated through numerical tests

dynamics of the formation of two dimensional ordered structures the growth of ordered domains in lattice gas models which occurs after the system is quenched from infinite temperature to a state below the critical temperaturet c is studied by monte carlo simulation for a square lattice with repulsion between nearest and next nearest neighbors which in equilibrium exhibits fourfold degenerate 2 1 superstructures the time dependent energy e t domain size l t and structure functions q t are obtained both for glauber dynamics no conservation law and the case with conserved density kawasaki dynamics at late times the energy excess and halfwidth of the structure factor decrease proportional tot x whilel t t x where the exponent x 1 2 for glauber dynamics and x 1 3 for kawasaki dynamics in addition the structure factor satisfies a scaling laws k t t 2xs ktx the smaller exponent for the conserved density case is traced back to the excess density contained in the walls between ordered domains which must be redistributed during growth quenches tot t c t tc where we estimate dynamic critical exponents andt 0 are also considered in the latter case the system becomes frozen in a glasslike domain pattern far from equilibrium when using kawasaki dynamics the generalization of our results to other lattices and structures also is briefly discussed

the center functor is fully faithful abstract we prove that the notion of drinfeld center defines a functor from the category of indecomposable multi tensor categories with morphisms given by bimodules to that of braided tensor categories with morphisms given by monoidal bimodules moreover we apply some ideas from the physics of topological orders to prove that the center functor restricted to indecom posable multi fusion categories with additional conditions on the target category is fully faithful as byproducts we provide new proofs to some important known results in fusion categories in physics this fully faithful functor gives the precise mathematical description of the boundary bulk relation for 2 1d anomaly free topological orders with gapped boundaries

weakly group theoretical and solvable fusion categories we introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups weakly group theoretical categories and solvable categories these are fusion categories that are morita equivalent to iterated extensions in the world of fusion categories of arbitrary respectively solvable finite groups weakly group theoretical categories have integer dimension and all known fusion categories of integer dimension are weakly group theoretical our main results are that a weakly group theoretical category c has the strong frobenius property i e the dimension of any simple object in an indecomposable c module category divides the dimension of c and that any fusion category whose dimension has at most two prime divisors is solvable a categorical analog of burnside s theorem for finite groups this has powerful applications to classification of fusion categories and semsisimple hopf algebras of a given dimension in particular we show that any fusion category of integer dimension 84 is weakly group theoretical i e comes from finite group theory and give a full classification of semisimple hopf algebras of dimensions pqr and pq2 where p q r are distinct primes

an explicit hybridized discontinuous galerkin method for serre green naghdi wave model abstract simulation of coastal water waves requires a relevant mathematical model and a proper numerical technique which can be used in irregular geometries in this paper we study the numerical solution to a class of irrotational green naghdi gn equation using the hybridized discontinuous galerkin method hdg we first use strang splitting to decompose this problem into a hyperbolic and a dispersive sub problem for the hyperbolic sub problem we use an explicit hdg with lax friedrichs numerical fluxes we also include different types of boundary conditions in our formulation similar to other hdg implementations our global unknowns are the numerical traces i e the unknowns on the mesh skeleton this choice can significantly reduce the number of global degrees of freedom solving this part of the problem involves newton iterations at each time step however unlike the implicit versions of hdg the element equations are decoupled and we do not need to solve a large global system of equations on the other hand for the dispersive sub problem we need to solve a global system of equations implicitly since hdg only involves unknowns on the mesh skeleton it can be a very efficient alternative for this implicit solve step by coupling these two solvers we obtain the solution to the green naghdi equation next we use this technique to solve a set of verification and validation examples in the first example we show the convergence properties of the numerical method next we compare our results with a set of experimental data for the nonlinear water waves in different situations finally we present a two dimensional simulation result for a solitary wave passing over a cosine bump we observe close to optimal convergence rates and a good agreement between our numerical results and the experimental data

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

discontinuous galerkin methods for anisotropic semidefinite diffusion with advection we construct and analyze a discontinuous galerkin method to solve advection diffusion reaction pdes with anisotropic and semidefinite diffusion the method is designed to automatically detect the so called elliptic hyperbolic interface on fitted meshes the key idea is to use consistent weighted average and jump operators optimal estimates in the broken graph norm are proven these are consistent with well known results when the problem is either hyperbolic or uniformly elliptic the theoretical results are supported by numerical evidence

a gpu based multi level algorithm for boundary value problems abstract a novel geometric single grid multi level gsgml algorithm is presented for the numerical solution of boundary value problems bvps the algorithm is specifically designed for parallel scalability with high occupancy of streaming processors inside many core architectures like the graphics processing unit gpu the algorithm consists of iterative superposed operations on a single grid and is composed of two full grid routines a restriction and a coarsened interpolation relaxation the restriction is a local average of scalar or vector fields and the interpolation relaxation is a simultaneous application of coarsened finite difference operators and interpolations in complementary subsets of the grid the routines are scheduled in a saw like refining cycle convergence to machine precision is achieved repeating the full cycle using residuals and superposition the algorithm is attractive for its computational simplicity and provides a competitive gpu based fast solver for elliptic partial differential equations like the poisson helmholtz equation applications shown in this work include the deformation of two dimensional grids the computation of three dimensional streamlines for a singular trifoil knot vortex and the calculation of three dimensional electric potentials in heterogeneous dielectric media

geometric properties of random disk packings random packings ofn 2000 rigid disks in the plane subject to periodic boundary conditions on a square primitive cell have been generated by a concurrent construction which treats all disks on an equal footing as opposed to previously investigated sequential constructions the particles start with random positions and velocities and as they move about they grow uniformly in size from points to jammed disks the collection of packings displays several striking geometric features these include for largen typically polycrystalline textures with irregular grain boundaries and linear shear fractures the packings occasionally contain monovacancies and trapped but unjammed rattler disks the latter appear to be confined to the grain boundaries the linear shear fractures preserve bond orientational order but disrupt translational order within the crystalline grains a new efficient event driven simulation algorithm is employed to generate the histories of colliding and jamming disks on a computer which can process one million floating point instructions per second the algorithm processes more than one million pairwise collisions per hour

error estimates to smooth solutions of runge kutta discontinuous galerkin methods for scalar conservation laws in this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for runge kutta discontinuous galerkin rkdg methods where the time discretization is the second order explicit total variation diminishing tvd runge kutta method error estimates for the mathbb p 1 piecewise linear elements are obtained under the usual cfl condition tau leq gamma h for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions where h and tau are the maximum element lengths and time steps respectively and the positive constant gamma is independent of h and tau however error estimates for higher order mathbb p k k geq 2 elements need a more restrictive time step tau leq gamma h 4 3 we remark that this stronger condition is indeed necessary as the method is linearly unstable under the usual cfl condition tau leq gamma h for the mathbb p k elements of degree k geq 2 error estimates of o h k 1 2 tau 2 are obtained for general monotone numerical fluxes and optimal error estimates of o h k 1 tau 2 are obtained for upwind numerical fluxes

on a cell entropy inequality for discontinuous galerkin methods we prove a cell entropy inequality for a class of high order dis continuous galerkin finite element methods approximating conservation laws which implies convergence for the one dimensional scalar convex case

locally frozen defects in random sequential adsorption with diffusional relaxation abstract random sequential adsorption with diffusional relaxation of two by two square objects on the two dimensional square lattice is studied by monte carlo computer simulation asymptotically for large lattice sizes diffusional relaxation allows the deposition process to reach full coverage the coverage approaches the full occupation value 1 as a power law with convergence exponent near 1 2 for a periodic lattice of finite even size l the final state is a frozen random rectangular grid of domain walls connecting single site defects the domain sizes saturate at l0 8 prior to saturation i e asymptotically for infinite lattice the domain growth is power law with growth exponent near or possibly somewhat smaller than 1 2

a splitting approach for the fully nonlinear and weakly dispersive green naghdi model the fully nonlinear and weakly dispersive green naghdi model for shallow water waves of large amplitude is studied the original model is first recast under a new formulation more suitable for numerical resolution an hybrid finite volume and finite difference splitting approach is then proposed which could be adapted to many physical models that are dispersive corrections of hyperbolic systems the hyperbolic part of the equations is handled with a high order finite volume scheme allowing for breaking waves and dry areas the dispersive part is treated with a classical finite difference approach extensive numerical validations are then performed in one horizontal dimension relying both on analytical solutions and experimental data the results show that our approach gives a good account of all the processes of wave transformation in coastal areas shoaling wave breaking and run up

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

a quadtree adaptive multigrid solver for the serre green naghdi equations the serre green naghdi sgn equations also known as the fully nonlinear boussinesq wave equations accurately describe the behaviour of dispersive shoaling water waves this article presents and validates a novel combination of methods for the numerical approximation of solutions to the sgn equations the approach preserves the robustness of the original finite volume saint venant solver in particular for the treatment of wetting drying and equilibrium states the linear system of coupled vector equations governing the dispersive sgn momentum sources is solved simply and efficiently using a generic multigrid solver this approach generalises automatically to adaptive quadtree meshes adaptive mesh refinement is shown to provide orders of magnitude gains in speed and memory when applied to the dispersive propagation of waves during the tohoku tsunami the source code test cases and examples are freely available

gerris a tree based adaptive solver for the incompressible euler equations in complex geometries an adaptive mesh projection method for the time dependent incompressible euler equations is presented the domain is spatially discretised using quad octrees and a multilevel poisson solver is used to obtain the pressure complex solid boundaries are represented using a volume of fluid approach second order convergence in space and time is demonstrated on regular statically and dynamically refined grids the quad octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features the source code of the method implementation is freely available

weighted interior penalty discretization of fully nonlinear and weakly dispersive free surface shallow water flows in this paper we further investigate the use of a fully discontinuous finite element discrete formulation for the study of shallow water free surface flows in the fully nonlinear and weakly dis persive flow regime we consider a decoupling strategy in which we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non hydrostatic effects this source term can be computed through the resolution of elliptic second order linear sub problems which only involve second order partial derivatives in space we then introduce an associated symmetric weighted internal penalty discrete bilinear form allowing to deal with the discontinuous nature of the elliptic problem s coefficients in a stable and consistant way similar discrete formulations are also introduced for several recent optimized fully nonlinear and weakly dis persive models these formulations are validated again several benchmarks involving h convergence p convergence and comparisons with experimental data showing optimal convergence properties

large time existence for 3d water waves and asymptotics we rigorously justify in 3d the main asymptotic models used in coastal oceanography including shallow water equations boussinesq systems kadomtsev petviashvili kp approximation green naghdi equations serre approximation and full dispersion model we first introduce a variable nondimensionalized version of the water waves equations which vary from shallow to deep water and which involves four dimensionless parameters using a nonlocal energy adapted to the equations we can prove a well posedness theorem uniformly with respect to all the parameters its validity ranges therefore from shallow to deep water from small to large surface and bottom variations and from fully to weakly transverse waves

monoidal 2 structure of bimodule categories abstract we define a notion of tensor product of bimodule categories and prove that with this product the 2 category of c bimodule categories for fixed tensor c is a monoidal 2 category in the sense of kapranov and voevodsky 1991 1 we then provide a monoidal structure preserving 2 equivalence between the 2 category of c bimodule categories and z c module categories module categories over the center of c for a finite group g we show that de equivariantization is equivalent to the tensor product over rep g we derive rep g module fusion rules and show that the group of invertible rep g module categories is isomorphic to h 2 g k extending results in etingof et al 2

boussinesq type formulations for fully nonlinear and extremely dispersive water waves derivation and analysis boussinesq formulations valid for highly dispersive and highly nonlinear water waves are derived with the objective of improving the accuracy of the vertical variation of the velocity field as well

discontinuous galerkin methods for solving boussinesq green naghdi equations in resolving non linear and dispersive surface water waves a local discontinuous galerkin method for boussinesq green naghdi equations is presented and validated against experimental results for wave transformation over a submerged shoal currently green naghdi equations have many variants in this paper a numerical method in one dimension is presented for the green naghdi equations based on rotational characteristics in the velocity field stability criterion is also established for the linearized green naghdi equations for both the analytical problem and the numerical method verification is done against a linearized standing wave problem in flat bathymetry and h p denoted by k in this paper error rates are plotted validation plots show good agreement of the numerical results with the experimental ones

random sequential adsorption series expansion and monte carlo simulation abstract random sequential adsorption is an irreversible surface deposition of extended objects in systems with continuous degrees of freedom coverage follows a power law t j ct where the exponent depends on the geometric shape symmetry of the objects lattice models give typically exponential saturation to jamming coverage we discuss how such function t can be computed by series expansions and analyzed with pade approximations we consider the applications of efficient monte carlo computer simulation method event driven method to random sequential adsorptions with high precision and at very long time scale

a well balanced element free galerkin method for the nonlinear shallow water equations abstract in this paper we consider the nonlinear shallow water equations over variable bottom topography in one dimension and propose a well balanced element free galerkin method for solving this system the proposed scheme has the features of being high order accurate for general solutions and exactly preserving the still water stationary solution the main ingredient to achieve the well balanced property is to use a special decomposition to the source term and discretize the source term as the flux term numerical tests are presented to illustrate the accuracy and validity of the proposed scheme

functoriality of the center of an algebra the notion of the center of an algebra over a field k has a far reaching generalization to algebras in monoidal categories the center then lives in the monoidal center of the original category in this paper we study functorial properties of the center we show that it gives rise to a 2 functor from the bicategory of semisimple indecomposable module categories over a fusion category to the bicategory of commutative algebras in the monoidal center of this fusion category morphism spaces of the latter bicategory are extended from algebra homomorphisms to certain categories of cospans we conjecture that the above 2 functor arises from a lax 3 functor between tricategories and that in this setting one can relax the conditions from fusion categories to finite tensor categories our construction is motivated by two dimensional conformal field theory enriched by defects of all codimensions

high order well balanced cdg fe methods for shallow water waves by a green naghdi model in this paper we consider a one dimensional fully nonlinear weakly dispersive green naghdi model for shallow water waves over variable bottom topographies such model describes a large spectrum of shallow water waves and it is thus of great importance to design accurate and robust numerical methods for solving it the governing equations contain mixed spatial and temporal derivatives of the unknowns they also have still water stationary solutions which should be preserved in stable numerical simulations in our numerical approach we first reformulate the green naghdi equations into balance laws coupled with an elliptic equation we then propose a family of high order numerical methods which discretize the balance laws with well balanced central discontinuous galerkin methods and the elliptic part with continuous finite element methods linear dispersion analysis for both the reformulated green naghdi system and versions of the proposed numerical scheme is performed when the bottom topography is flat numerical tests are presented to illustrate the accuracy and stability of the proposed schemes as well as the capability of the green naghdi model to describe a wide range of shallow water wave phenomena

long time effects of bottom topography in shallow water abstract we present and discuss new shallow water equations that provide an estimate of the long time asymptotic effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid with a free surface which is moving under the force of gravity we consider the regime where the froude number is much smaller than the aspect ratio of the shallow domain the new equations are obtained at first order in an asymptotic expansion of the solutions of the euler equations for a shallow fluid by using the small parameter 2 the leading order equations in this expansion enforce hydrostatic balance while those obtained at first order retain certain nonhydrostatic effects both sets of equations conserve energy and circulation convect potential vorticity and have a hamiltonian formulation the corresponding energy and enstrophy are quadratic integrals with which we can bound the cumulative influence of the nonhydrostatic effects

discrete functional analysis tools for discontinuous galerkin methods with application to the incompressible navier stokes equations two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes i a discrete counterpart of the continuous sobolev embeddings in both hilbertian and non hilbertian settings ii a compactness result for bounded sequences in a suitable discontinuous galerkin norm together with a weak convergence property for some discrete gradients the proofs rely on techniques inspired by the finite volume literature which differ from those commonly used in finite element analysis the discrete functional analysis tools are used to prove the convergence of discontinuous galerkin approximations of the steady incompressible navier stokes equations two discrete convective trilinear forms are proposed a non conservative one relying on temam s device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure

high order well balanced central local discontinuous galerkin finite element methods for solving the green naghdi model abstract in this paper a hybrid numerical method combining the central local discontinuous galerkin method with continuous finite element method is proposed to solve the fully nonlinear weakly dispersive green naghdi model describing a large spectrum of shallow water waves in our numerical approach the green naghdi model is first rewritten as balance laws coupled with an elliptic equation in terms of new variables adapted for numerical studies then we discretize the balance laws with well balanced central local discontinuous galerkin methods and the elliptic part with continuous finite element methods numerical tests are presented to illustrate the performance of the proposed schemes