High Order Accuracy

For the numerical solution of initial value problems a general procedure to determine global integration methods is derived and studied. They are collocation methods which can be easily implemented and provide a high order accuracy. They further provide globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients are generated by a recurrence formula and no integrals are involved in the calculation. Numerical experiments provide favorable comparisons with other existing methods.

A finite volume hybrid scheme for the spatial discretization that combines a fixed stencil and a stencil determined by the classical essentially non-oscillatory (ENO) scheme is presented. Evolution equations are obtained for the mean values of each cell by means of piecewise interpolation. Time discretization is accomplished by a classical fourth-order Runge-Kutta. Interpolation polynomials are determined using information of adjacent cells. While smooth regions are interpolated by means of a fixed molecule, discontinuous or sharp regions are interpolated by the classical ENO algorithm. The algorithm estimates the interpolation error at each time step by means of two interpolants of order q and q +1. The main computational load of the resultant scheme is in the interpolation, which is performed by the divided differences table. This table involves O(qN) operations, where q is the interpolation order and N is the number of cells. Finally, linear test cases of continuous and discontinuous initial conditions are integrated to see the goodness of the hybrid scheme. It is well known that, for some particular initial conditions, the classical ENO scheme does not perform properly, not attaining the truncation error of the scheme. It is shown that, for the smooth initial condition, sin 4 (x), the classical ENO scheme does not preserve the character of stability of the initial value problem, giving rise to unstable eigenvalues. The proposed hybrid scheme solves this problem, choosing a fixed stencil over the whole computational domain. The resultant schemes are equivalent to the classical finite difference schemes, which preserve the character of stability. It is also known that the same degeneracy of the error can be encountered for discontinuous solutions. It is shown for the initial discontinuous solution, e -x , that the classical ENO algorithm does not perform properly due to the conflict between the selection of the stencil to smoother regions (downwind region) and the hyperbolic character of the problem, which obliges us to take information from downwind. The proposed hybrid scheme solves this problem by choosing a fixed stencil over the whole computational domain except at the discontinuity.

In this short note we address the issue of numerical resolution and efficiency of high order weighted essentially nonoscillatory (WENO) schemes for computing solutions containing both discontinuities and complex solution features, through two representative numerical examples: the double Mach reflection problem and the Rayleigh-Taylor instability problem. We conclude that for such solutions with both discontinuities and complex solution features, it is more economical in CPU time to use higher order WENO schemes to obtain comparable numerical resolution.

This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-water well-balanced schemes for the shallow water equations. We concentrate on numerical examples with solutions near a moving-water equilibrium. For such examples, still-water well-balanced methods are not capable of capturing the small perturbations of the moving-water equilibrium and may generate significant spurious oscillations, unless an extremely refined mesh is used. On the other hand, movingwater well-balanced methods perform well in these tests. The numerical examples in this note clearly demonstrate the importance of utilizing moving-water well-balanced methods for solutions near a moving-water equilibrium.

A conservative 3-D discontinuous Galerkin (DG) baroclinic model has been developed in the NCAR High-Order Method Modeling Environment (HOMME) to investigate global atmospheric flows. The computational domain is a cubed-sphere free from coordinate singularities. The DG discretization uses a high-order nodal basis set of orthogonal Lagrange-Legendre polynomials and fluxes of inter-element boundaries are approximated with Lax-Friedrichs numerical flux. The vertical discretization follows the 1-D vertical Lagrangian coordinates approach combined with the cell-integrated semi-Lagrangian method to preserve conservative remapping. Time integration follows the third-order SSP-RK scheme. To valid proposed 3-D DG model, the baroclinic instability test suite proposed by Jablonowski and Williamson is investigated. Parallel performance is evaluated on IBM Blue Gene/L and IBM POWER5 p575 supercomputers. * This paper is supported by the DOE-SciDAC program under award #DE-FG02-04ER63870.

We present a fast Poisson solver on spherical shells. With a special change of variable, the radial part of the Laplacian transforms to a constant coefficient differential operator. As a result, the Fast Fourier Transform can be applied to solve the Poisson equation with O(N 3 log N) operations. Numerical examples have confirmed the accuracy and robustness of the new scheme.

We present a fast Poisson solver on spherical shells. With a special change of variable, the radial part of the Laplacian transforms to a constant coefficient differential operator. As a result, the Fast Fourier Transform can be applied to solve the Poisson equation with operations. Numerical examples have confirmed the accuracy and robustness of the new scheme.

This paper evaluates the use of the compressible Navier-Stokes equations, with prescribed zero velocities, as a model for heat transfer in solids. In particular in connection with conjugate heat transfer problems. We derive estimates, and show how to choose and scale the coefficients of the energy part in the Navier-Stokes equations, such that the difference between the energy equation and the heat equation is minimal. A rigorous analysis of the physical interface conditions for the conjugate heat transfer problem is performed and energy estimates are derived in non-standard L 2equivalent norms. The numerical schemes are proven energy stable with the physical interface conditions and the stability of the schemes are independent of the order of accuracy. We have performed computations of a conjugate heat transfer problem in two different ways. One where, as traditionally, the heat transfer in the solid is governed by the heat equation. The other where the heat transfer in the solid is governed by the Navier-Stokes equations. The simulations are compared and the numerical results corroborate the theoretical results.

A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.

In this article, recent developments for increased performance of the high order and stable SBP-SAT finite difference technique is described. In particular we discuss the use of weak boundary conditions and dual consistent formulations. The use of weak boundary conditions focus on increased convergence to steady state, and hence efficiency. Dual consistent schemes produces superconvergent functionals and increases accuracy.

In this article, well-posedness and dual consistency of the linearized incompressible Navier-Stokes equations posed on time-dependent spatial domains are studied. To simplify the derivation of the dual problem, the second order formulation is transformed to first order form. Boundary conditions that simultaneously lead to well-posedness of the primal and dual problems are derived. We construct fully discrete finite difference schemes on summation-byparts form, in combination with the simultaneous approximation technique. We prove energy stability and discrete dual consistency. Moreover, we show how to construct the penalty operators such that the scheme automatically adjusts to the variations of the spatial domain, and as a result, stability and discrete dual consistency follow simultaneously. The method is illustrated by considering a deforming time-dependent spatial domain in two dimensions. The numerical calculations are performed using high order operators in space and time. The results corroborate the stability of the scheme and the accuracy of the solution. We also show that linear functionals are superconverging. Additionally, we investigate the convergence of non-linear functionals and the divergence of the solution.

In this article we consider a nonlocal problem with the second kind integral condition for a parabolic equation. Under some conditions on initial data we proved existence and uniqueness of a generalized solution applying the method of a priori estimates and a parameter continuation method.

We prove the existence and uniqueness of a strong solution for a parabolic singular equation in which we combine Dirichlet with integral boundary conditions given only on parts of the boundary. The proof uses a priori estimate and the density of the range of the operator generated by the problem considered.

8208], we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous Galerkin (DG) finite element solver and a first order finite difference solver for Eikonal equations. The method has second order accuracy in the L 1 norm and a very fast convergence speed, but only first order accuracy in the L ∞ norm for the general cases. This is an obstacle to the design of higher order DG fast sweeping methods. In this paper, we overcome this problem by developing uniformly accurate DG fast sweeping methods for solving Eikonal equations. We design novel causality indicators which guide the information flow directions for the DG local solver. The values of these indicators are initially provided by the first order finite difference fast sweeping method, and they are updated during iterations along with the solution. We observe both a uniform second order accuracy in the L ∞ norm (in smooth regions) and the fast convergence speed (linear computational complexity) in the numerical examples.

In this paper we prove stability of Robin solid wall boundary conditions for the compressible Navier-Stokes equations. Applications include the no-slip boundary conditions with prescribed temperature or temperature gradient and the first order slip-flow boundary conditions. The formulation is uniform and the transitions between different boundary conditions are done by a change of parameters. We give different sharp energy estimates depending on the choice of parameters. The discretization is done using finite differences on Summation-By-Parts form with weak boundary conditions using the Simultaneous Approximation Term. We verify convergence by the method of manufactured solutions and show computations of flows ranging from no-slip to substantial slip.

In this paper we prove stability of Robin solid wall boundary conditions for the compressible Navier-Stokes equations. Applications include the no-slip boundary conditions with prescribed temperature or temperature gradient and the first order slip-flow boundary conditions. The formulation is uniform and the transitions between different boundary conditions are done by a change of parameters. We give different sharp energy estimates depending on the choice of parameters. The discretization is done using finite differences on Summation-By-Parts form with weak boundary conditions using the Simultaneous Approximation Term. We verify convergence by the method of manufactured solutions and show computations of flows ranging from no-slip to substantial slip.

In this paper we prove stability of Robin solid wall boundary conditions for the compressible Navier-Stokes equations. Applications include the no-slip boundary conditions with prescribed temperature or temperature gradient and the first order slip-flow boundary conditions. The formulation is uniform and the transitions between different boundary conditions are done by a change of parameters. We give different sharp energy estimates depending on the choice of parameters. The discretization is done using finite differences on Summation-By-Parts form with weak boundary conditions using the Simultaneous Approximation Term. We verify convergence by the method of manufactured solutions and show computations of flows ranging from no-slip to substantial slip.

In this paper we construct a hierarchy of arbitrary high (even) order accurate explicit time propagators for semi-discrete second order hyperbolic systems. An accurate semi-discrete problem is obtained by approximating the corresponding spatial derivatives using high order accurate finite difference operators satisfying the summation by parts rule. In order to obtain a strictly stable semi-discrete problem, boundary conditions are imposed weakly using the simultaneous approximation term method. The time discretization starts with a second order central difference scheme, then using the modified equation approach (even in the presence of a first order derivative in time) we derive arbitrary high order accurate time marching schemes. For the fully discrete problem, we introduce a suitable weighted inner product and use the energy method to derive an optimal CFL condition, which provides a useful and rigorous criterion for stability. Numerical examples are also provided.

At the late stage of transitional boundary layers, the nonlinear evolution of the ring-like vortices and spike structures and their effects on the surrounding flow were studied by means of direct numerical simulation with high order accuracy. A spatial transition of the flat-plate boundary layers in the compressible flow was conducted. Detailed numerical results with high resolution clearly represented the typical vortex structures, such as ring-like vortices and so on, and induced ejection and sweep events. It was verified that the formation of spike structures in transitional boundary layers had close relationship with ring-like vortices. Especially, compared to the newly observed positive spike structure in the experiments, the same structure was found in the present numerical simulations, and the mechanism was also studied and analyzed.

We discuss the basics of discontinuous Galerkin methods (DG) for CEM as an alternative of emerging importance to the widely used FDTD. The benefits of DG methods include geometric flexibility, high-order accuracy, explicit timeadvancement, and very high parallel performance for large scale applications. The performance of the scheme shall be illustrated by several examples. As an example of particular interest, we further explore efficient probabilistic ways of dealing with uncertainty and uncertainty quantification in electromagnetics applications. Whereas the discussion often draws on scattering applications, the techniques are applicable to general problems in CEM.

Title of program:RPROP Nature of physical problem Coupled second-order differential equations which arise in Catalogue number: AAJK electron collision with atoms, ions and molecules are solved over a given range of the independent variable. The R-matrix at Program obtainable from: CPC Program Library, Queen's Uni-one end of the range is calculated given the R-matrix at the versity, Belfast, N. Ireland (see application form in this issue) other end of the range.

Ever since its introduction by Kane Yee over forty years ago, the finite-difference time-domain (FDTD) method has been a widely-used technique for solving the time-dependent Maxwell's equations. This paper presents an alternative approach to these equations in the case of spatially-varying electric permittivity and/or magnetic permeability, based on Krylov subspace spectral (KSS) methods. These methods have previously been applied to the variablecoefficient heat equation and wave equation, and have demonstrated high-order accuracy, as well as stability characteristic of implicit time-stepping schemes, even though KSS methods are explicit. KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Gene Golub and Gérard Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral, rather than physical, domain. We show how they can be generalized to coupled systems of equations, such as Maxwell's equations, by choosing appropriate basis functions that, while induced by this coupling, still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields. We also discuss the implementation of appropriate boundary conditions for simulation on infinite computational domains, and how discontinuous coefficients can be handled.

This paper describes a parallel implementation of the discontinuous Galerkin method. The discontinuous Galerkin is a spatially compact method that retains its accuracy and robustness on non-smooth unstructured grids and is well suited for time dependent simulations. Several parallelization approaches are studied and evaluated. The most natural and symmetric of the approaches has been implemented in an object-oriented code used to simulate aeroacoustic scattering. The parallel implementation is MPI-based and has been tested on various parallel platforms such as the SGI Origin, IBM SP2, and clusters of SGI and Sun workstations. The scalability results presented for the SGI Origin show slightly superlinear speedup on a fixed-size problem due to cache effects.

The objective of this paper is to present a new Large Eddy Simulation (LES) model obtained by filtering a generalized version of the Navier-Stokes equations with nonlinear viscosity. This new model is a generalization of the model introduced in [J. M. Rodríguez, R. Taboada-Vázquez, A new LES model derived from generalized Navier-Stokes equations with nonlinear viscosity. Computers and Mathematics with Applications 73 (2017) 294-303]. The new LES model, in which viscosity has been substituted by a non linear function of the strain rate tensor, has been implemented and validated using FreeFem++ codes. The numerical predictions have been compared with analytical solutions of the Navier-Stokes equations and it was found that the present model provides a more accurate approximation of the exact solution than that obtained by Direct Numerical Simulation (DNS) of the Navier-Stokes equations, even using a coarser mesh. The model has also been validated by studying the unsteady flow over a backwardfacing step. In this case, our numerical predictions have been compared with the experimental measurements reported by Armaly et al. and the numerical results obtained by Chacón and Lewandowski. The new model performs satisfactorily in predicting these flows too.

There is a growing interest in using mathematical models to understand crime dynamics, crime prevention, and detection. The past decade has experienced a relative reduction in conventional crimes, but this has been replaced by significant increases in cybercrime. We use a system of nonlinear differential equations to explore the impact of criminal location in relation to cyber networks on the amount of cybercrime. Using steady-state analysis and extensive numerical simulations, we find that the location of criminals relative to the network does not impact the system qualitatively although there are quantitative differences. Cyber networks that are more clustered are likely to experience greater levels of cybercrime but there is also a saturation effect that limits the level of victimisation as the number of criminals attempting to undertake crimes on a given network increases.

We introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability is analysed for continuous and discrete problems using both strong and weak formulations, and a general transmission condition is obtained. The theory is applied to several examples including the coupling of fluid flow models, multi-grid implementations, multi-block formulations and numerical filtering.

A stencil-adaptive SBP-SAT finite difference scheme is shown to display superconvergent behavior. Applied to the linear advection equation, it has a convergence rate O(∆x 4) in contrast to a conventional scheme, which converges at a rate O(∆x 3).

We extend the construction of so-called encapsulated global summation-by-parts operators to the general case of a mesh which is not boundary conforming. Owing to this development, energy stable discretizations of nonlinear and variable coefficient initial boundary value problems can be formulated in simple and straightforward ways using high-order accurate operators of generalized summation-by-parts type. Encapsulated features on a single computational block or element may include polynomial bases, tensor products as well as curvilinear coordinate transformations. Moreover, through the use of inner product preserving interpolation or projection, the global summation-by-parts property is extended to arbitrary multi-block or multi-element meshes with non-conforming nodal interfaces.

The discontinuous Galerkin (DG) method continues to maintain heightened levels of interest within the simulation community because of the discretization flexibility it provides. One of the fundamental properties of the DG methodology and arguably its most powerful property is the ability to combine high-order discretizations on an inter-element level while allowing discontinuities between elements. This flexibility, however, generates a plethora of difficulties when one attempts to use DG fields for feature extraction and visualization, as most post-processing schemes are not designed for handling explicitly discontinuous fields. This work introduces a new method of applying smoothness-increasing, accuracy-conserving filtering on discontinuous Galerkin vector fields for the purpose of enhancing streamline integration. The filtering discussed in this paper enhances the smoothness of the field and eliminates the discontinuity between elements, thus resulting in more accurate streamlin...

We present a fast direct solver for two dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of the Lippmann-Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require O(N 3/2) work, where N denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both the low and high frequency regimes.

A spectral element (SE) implementation of the Givoli-Neta non-reflecting boundary condition (NRBC) is considered for the solution of the Klein-Gordon equation. The infinite domain is truncated via an artificial boundary B, and a high-order NRBC is applied on B. Numerical examples, in various configurations, concerning the propagation of a pressure pulse are used to demonstrate the performance of the SE implementation. Effects of time integration techniques and long term results are discussed. Specifically, we show that in order to achieve the full benefits of high-order accuracy requires balancing all errors involved; this includes the order of accuracy of the spatial discretization method, timeintegrators, and boundary conditions.

A reduced shallow water model under constant, non-zero advection in the infinite channel is considered. High-order (Givoli-Neta) non-reflecting boundary conditions are introduced in various configurations to create a finite computational space and solved using a spectral element formulation with high-order time integration. Numerical examples are used to demonstrate the synergy of using high-order spatial, time, and boundary discretization. We show that by balancing all numerical errors involved, high-order accuracy can be achieved for unbounded domain problems.

A stencil-adaptive SBP-SAT finite difference scheme is shown to display superconvergent behavior. Applied to the linear advection equation, it has a convergence rate O(∆x 4) in contrast to a conventional scheme, which converges at a rate O(∆x 3).

A wide range of applications requires an accurate solution of a particular Hamilton-Jacobi (H-J) equation known as the Eikonal equation. In this paper, we employ the Chebyshev pseudospectral viscosity method to solve this equation. This method essentially consists of adding a spectral viscosity to the equation for high wave numbers of the numerical solution. This spectral viscosity, which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme. Here the method is described in detail and the numerical results for several examples are provided which reveals the efficiency of the proposed method.

Fixed-point iterative sweeping methods were developed in the literature to efficiently solve steady state solutions of Hamilton-Jacobi equations and hyperbolic conservation laws. Similar as other fast sweeping schemes, the key components of this class of methods are the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. A family of characteristics of the corresponding hyperbolic partial differential equations (PDEs) are covered in a certain direction simultaneously in each sweeping order. Furthermore, good properties of fixed-point iterative sweeping methods which are different from other fast sweeping methods include that they have explicit forms and do not involve inverse operation of nonlinear local systems, and they can be applied to general hyperbolic equations using any monotone numerical fluxes and high order approximations easily. In a recent article [L. Wu, Y.-T. Zhang, S. Zhang and C.-W. Shu, Commun. Comput. Phys., 20 (2016), 835-869], a fifth order fixed-point sweeping WENO scheme was designed for solving steady state of hyperbolic conservation laws, and it was shown that the scheme converges to steady state solution much faster than the regular total variation diminishing (TVD) Runge-Kutta time-marching approach by stability improvement of high order schemes with a forward Euler time-marching. An open problem is that for some benchmark numerical examples, the iteration residue of the fixed-point sweeping WENO scheme hangs at a truncation error level instead of settling down to machine zero. This issue makes it difficult to determine the convergence criterion for the iteration and challenging to apply the method to complex problems. To solve this issue, in this paper we apply the multi-resolution WENO scheme developed in [J. Zhu and C.-W. Shu, J. Comput. Phys., 375 (2018), 659-683] to the fifth order fixed-point sweeping WENO scheme and obtain an absolutely convergent fixedpoint fast sweeping method for steady state of hyperbolic conservation laws, i.e., the residue of the fast sweeping iterations converges to machine zero / round off errors for all benchmark problems tested. Extensive numerical experiments, including solving difficult problems such as the shock reflection and supersonic flow past plates, are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth order sweeping scheme.

Classical longitude-latitude grids result in singularities at the two poles, where the meridians converge. Dealing with those singularities requires the use of filtering with operators, which are non-local and thus significantly reduce the efficiency. This difficulty has recently renewed the interest for numerical work on the idea of using Cartesian grids on sphere. This work focuses on building a systematic method to generate Cartesian grid over sphere, developing the necessary mathematical formulation to solve the advection dominated scalar transport problems on the surface of sphere. In this work we present method of characteristics Galerkin method, which is high-order accurate, and highly efficient. The element level integrals can be analytically calculated and this fact contributes towards the computational efficiency of this method.

Radial basis function(RBF) methods have been actively developed in the last decades. The advantages of RBF methods are that these methods are mesh-free and yield high order accuracy if the function is smooth enough. The RBF approximation for discontinuous problems, however, deteriorates its high order accuracy due to the Gibbs phenomenon. With the Gibbs phenomenon in the RBF approximation, the L ∞ error remains only O(1). The main purpose of this paper is to show that high order accuracy can be recovered from the RBF approximation contaminated with the Gibbs phenomenon if a proper reconstruction method is applied. In this work, the Gegenbauer reconstruction method is used to reconstruct the RBF approximation for the recovery of high order accuracy. Several numerical examples presented in this work indicate that the Gegenbauer polynomials are Gibbs complementary to the RBF approximations and hence high order convergence can be recovered from the RBF approximations for discontinuous problems. These results also indicate that the Gegenbauer reconstruction method which was originally developed for the polynomial approximation works for non-polynomial basis methods such as the RBF method. Numerical examples including the linear and nonlinear hyperbolic partial differential equations are presented.

We use high order finite difference methods to solve the wave equation in the second order form. The spatial discretization is performed by finite difference operators satisfying a summation-by-parts property. The focus of this work is on the numerical treatment of non-conforming grid interfaces. The interface conditions are imposed weakly by the simultaneous approximation term technique in combination with interface operators, which move the discrete solutions between the grids on the interface. In particular, we consider interpolation operators and projection operators. A norm-compatibility condition, which leads to stability for first order hyperbolic systems, does not suffice for second order wave equations. An extra constraint on the interface operators must be satisfied to derive an energy estimate for stability. We carry out eigenvalue analyses to investigate the additional constraint and how it is related to stability, and find that the projection operators have better stability properties than the interpolation operators. In addition, a truncation error analysis is performed to study the convergence property of the numerical schemes. In the numerical experiments, the stability and accuracy properties of the numerical schemes are further explored, and the practical usefulness of non-conforming grid interfaces is presented and discussed in two efficiency studies.

A conservative 3-D discontinuous Galerkin (DG) baroclinic model has been developed in the NCAR High-Order Method Modeling Environment (HOMME) to investigate global atmospheric flows. The computational domain is a cubed-sphere free from coordinate singularities. The DG discretization uses a high-order nodal basis set of orthogonal Lagrange-Legendre polynomials and fluxes of inter-element boundaries are approximated with Lax-Friedrichs numerical flux. The vertical discretization follows the 1-D vertical Lagrangian coordinates approach combined with the cell-integrated semi-Lagrangian method to preserve conservative remapping. Time integration follows the third-order SSP-RK scheme. To valid proposed 3-D DG model, the baroclinic instability test suite proposed by Jablonowski and Williamson is investigated. Parallel performance is evaluated on IBM Blue Gene/L and IBM POWER5 p575 supercomputers.

Over the past three decades simplified empirical formulae contributed greatly in a rapid evaluation of the oil slick spreading and drifting. Modern oil spill models can utilise more accurate and physically relevant mathematical formulations. The suggested Multiphase Oil Spill Model is an attempt to take advantage of recent developments in areas of Computational Fluid Dynamics (CFD) and Environmental Modelling. A consistent Eulerian approach is applied across the model, the slick thickness is computed using layer-averaged Navier-Stokes equations, and the advection-diffusion equation is employed to simulate oil dynamics in the water column. To match the observed balance between advection, diffusion and spreading phenomena, a high-order accuracy numerical scheme is developed. Vertical dynamics of oil droplets plays a major role in oil mass exchange between the slick and the water column. Oil mixing by breaking waves is parameterised using newly developed kinetic equations. Majority parameters of oil, water column and breaking waves are conveniently combined into a single ''mixing factor'', quantifying partitioning of oil between the slick and the water column. The model is able to predict rate of oil entrainment for different scenarios of dispersant application with respect to the storm intensity and duration. Governing equations are verified using test cases, data and other models, and subsequently applied to Singapore Strait to simulate a hypothetical oil spill.

This paper presents an efficient time-domain method for computing the propagation of electromagnetic waves in microwave structures. The procedure uses high-order vector bases to achieve high-order accuracy in space, the Newmark's method to provide unconditional stability in time, and the transfinite-element method to truncate the waveguide ports. The resulting system matrix is real, symmetric, positive-definite, and can be solved by using the highly efficient multilevel preconditioned conjugate gradient algorithm. Since the method allows large time steps and nonuniform grids, the computational complexity for problems with irregular geometries is superior to the finite-difference time-domain method.

In CFD computations, discretization or truncation errors should be small providing an acceptable level of accuracy. In this paper, an extension is made of the recently proposed LES formalism based on sampling operators. It is shown that the sampling-based dynamic procedure, in combination with an appropriate truncation error model, can be used as a technique to increase the numerical accuracy of a discretization. The technique is resemblant to the well-known Richardson extrapolation. The procedure is tested on a 1D convection-diffusion equation and a 2D lid-driven cavity at Re = 400, using a finite difference method. Promising results are found.

In CFD computations, discretization errors or truncation errors should be small providing an acceptable level of accuracy. In this paper an alternative use is made of the recently proposed LES formalism based on sampling operators. It is shown that the sampling-based dynamic procedure, in combination with an appropriate truncation error model, can be used as a technique to increase the numerical accuracy of a discretization. The technique is resemblant to the well-known Richardson extrapolation. The procedure is tested on a 1D convection-diffusion equation and a 2D lid-driven cavity at Re = 400, using a finite difference method. Very promising results are obtained.

SummaryWe consider wave propagation in a coupled fluid‐solid region separated by a static but possibly curved interface. The wave propagation is modeled by the acoustic wave equation in terms of a velocity potential in the fluid, and the elastic wave equation for the displacement in the solid. At the fluid solid interface, we impose suitable interface conditions to couple the two equations. We use a recently developed energy‐based discontinuous Galerkin method to discretize the governing equations in space. Both energy conserving and upwind numerical fluxes are derived to impose the interface conditions. The highlights of the developed scheme include provable energy stability and high order accuracy. We present numerical experiments to illustrate the accuracy property and robustness of the developed scheme.

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Let −L be the generator of a Lévy semigroup on L 1 (R n) and f : R → R n be a nonlinearity. Nonlinear equations with nonlocal diffusive terms like ut + Lu + ∇ • f (u) = 0 appear as models with anomalous diffusion in continuum mechanics. We study the asymptotic behavior of solutions of these nonlocal equations as time t tends to infinity, analyzing the L p-decay and two terms of the asymptotics of solutions. In the critical case when the diffusion and nonlinear terms are balanced, i.e., L ∼ (−∆) α/2 , 1 < α < 2, f (s) ∼ s|s| (α−1)/n , the solutions feature genuinely nonlinear self-similar asymptotics. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Lois de conservation multifractales et de Lévy Résumé. Soit −L un générateur d'un semi-groupe de Lévy sur L 1 (R n) et soit f une fonction non linéaire. Les équations non linéaires avec des termes de diffusion non locaux du type ut + Lu + ∇ • f (u) = 0 servent de modèles avec « diffusion anomale » en mécanique des milieux continus. On trouve les taux de décroissance des normes L p (R n) des solutions ainsi que deux termes de l'asymptotique des solutions. Quand L ∼ (−∆) α/2 , 1 < α < 2, f (s) ∼ s|s| (α−1)/n , on montre que l'asymptotique temporelle est essentiellement non linéaire et déterminée par une solution auto-similaire. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Version française abrégée Nous nous intéressons au problème de Cauchy pour les équations non linéaires pseudo-différentielles, que nous appelons ici lois de conservation de Lévy u t + Lu + ∇ • f (u) = 0, u(x, 0) = u 0 (x), (0.1)

At the late stage of transitional boundary layers, the nonlinear evolution of the ring-like vortices and spike structures and their effects on the surrounding flow were studied by means of direct numerical simulation with high order accuracy. A spatial transition of the flat-plate boundary layers in the compressible flow was conducted. Detailed numerical results with high resolution clearly represented the typical vortex structures, such as ring-like vortices and so on, and induced ejection and sweep events. It was verified that the formation of spike structures in transitional boundary layers had close relationship with ring-like vortices. Especially, compared to the newly observed positive spike structure in the experiments, the same structure was found in the present numerical simulations, and the mechanism was also studied and analyzed.

In this paper, we present the hp-convergence analysis of a nondissipative high-order discontinuous Galerkin method on unstructured hexahedral meshes using a mass-lumping technique to solve the time-dependent Maxwell equations. In particular, we underline the spectral convergence of the method (in the sense that when the solutions and the data are very smooth, the discretization is of unlimited order). Moreover, we see that the choice of a non-standard approximate space (for a discontinuous formulation) with the absence of dissipation can imply a loss of spatial convergence. Finally we present a numerical result which seems to confirm this property.