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Abstract Next generation, conservative finite element hydrodynamic and transport models are vital for accurate and efficient ocean, estuary and river simulation. Numerical models such as these have been developed for decades by the U.S. Army Corps of Engineers at the Engineering Research and Development Center (ERDC). This paper focuses on recently developed implicit, multi-dimensional finite element 3D shallow water and transport models included in the Adaptive Hydraulics (AdH) numerical suite. These AdH 3D models benefit from their adaptive meshing capabilities to resolve sharp solution gradients, such as those often encountered with baroclinic wedges traveling up an estuary channel. This paper presents the AdH 3D mathematical formulation and solution procedure used to solve the weak finite element equations for these models, along with results for several common verification cases and an AdH Galveston Bay validation study. A novel Streamline Upwind Petrov–Galerkin (SUPG) method for 3D shallow water models is described which reduces to the AdH 2D shallow water SUPG formulation under certain conditions. Careful attention is placed on ensuring discrete consistency in the equation set. [ABSTRACT FROM AUTHOR]

The construction of local reduced-order models via multiscale basis functions has been an area of active research. In this paper, we propose online multiscale basis functions which are constructed using the offline space and the current residual. Online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators. We derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error. This error decrease is independent of physical parameters, such as the contrast and multiple scales in the problem. The offline spaces are constructed using Generalized Multiscale Finite Element Methods (GMsFEM). We show that if one chooses a sufficient number of offline basis functions, one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast. We note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects. Using the residual information, we can incorporate the distant information provided the offline approximation satisfies certain properties. In the paper, theoretical and numerical results are presented. Our numerical results show that if the offline space is sufficiently large (in terms of the dimension) such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues, then the error reduction by adding online multiscale basis function is independent of the contrast. We discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces. [ABSTRACT FROM AUTHOR]

Multidimensional Riemann solvers that have internal sub-structure in the strongly-interacting state have been formulated recently (D.S. Balsara (2012, 2014) [5,16] ). Any multidimensional Riemann solver operates at the grid vertices and takes as its input all the states from its surrounding elements. It yields as its output an approximation of the strongly interacting state, as well as the numerical fluxes. The multidimensional Riemann problem produces a self-similar strongly-interacting state which is the result of several one-dimensional Riemann problems interacting with each other. To compute this strongly interacting state and its higher order moments we propose the use of a Galerkin-type formulation to compute the strongly interacting state and its higher order moments in terms of similarity variables. The use of substructure in the Riemann problem reduces numerical dissipation and, therefore, allows a better preservation of flow structures, like contact and shear waves. In this second part of a series of papers we describe how this technique is extended to unstructured triangular meshes. All necessary details for a practical computer code implementation are discussed. In particular, we explicitly present all the issues related to computational geometry. Because these Riemann solvers are Mu ltidimensional and have S elf-similar strongly- I nteracting states that are obtained by C onsistency with the conservation law, we call them MuSIC Riemann solvers. (A video introduction to multidimensional Riemann solvers is available on http://www.nd.edu/~dbalsara/Numerical-PDE-Course .) The MuSIC framework is sufficiently general to handle general nonlinear systems of hyperbolic conservation laws in multiple space dimensions. It can also accommodate all self-similar one-dimensional Riemann solvers and subsequently produces a multidimensional version of the same. In this paper we focus on unstructured triangular meshes. As examples of different systems of conservation laws we consider the Euler equations of compressible gas dynamics as well as the equations of ideal magnetohydrodynamics (MHD). Several stringent test problems are solved for both PDE systems. [ABSTRACT FROM AUTHOR]

This paper concerns the stability of a class of Helmholtz problems in rectangular domains. A well known application is the electromagnetic scattering from a rectangular cavity embedded in an infinite ground plane. Error analysis of numerical methods for cavity problems relies heavily on the stability estimates. However, it is extremely difficult to derive an optimal stability bound with the explicit dependency on wave numbers. In this paper a high-order finite element approximation is proposed for calculating the stability bound. Numerical experiments show that the stability depends strongly on wave numbers in extreme case and it is almost independent on the wave numbers in an average sense. Our numerical results also help to understand the stability of the multi-frequency inverse problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we approximate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Hagedorn wave packets approach. By operator splitting, the Hamiltonian is divided into the modified part and the residual part. The modified Hamiltonian, which is the main new idea of this paper, is chosen by the fact that Hagedorn wave packets are localized both in space and momentum so that a crucial correction term is added to the truncated Hamiltonian, and is treated by evolving the parameters associated with the Hagedorn wave packets. The residual part is treated by a Galerkin approximation. We prove that, with the modified Hamiltonian only, the Hagedorn wave packets dynamics give the asymptotic solution with error , where ε is the scaled Planck constant. We also prove that, the Galerkin approximation for the residual Hamiltonian can reduce the approximation error to , where k depends on the number of Hagedorn wave packets added to the dynamics. This approach is easy to implement, and can be naturally extended to the multidimensional cases. Unlike the high order Gaussian beam method, in which the non-constant cut-off function is necessary and some extra error is introduced, the Hagedorn wave packets approach gives a practical way to improve accuracy even when ε is not very small. [Copyright &y& Elsevier]

Abstract: In this paper we present a new meshless method to solve well-known problems in physics and engineering with either constant or variable material properties in 2D space. Harmonic and bi-harmonic problems are considered in this paper. The method constructs a set of basis functions, called Equilibrated Basis Functions (EqBFs), through a weighted residual integration over a fictitious domain embedding the main one. The bases can satisfy the governing partial differential equations approximately. Chebyshev polynomials of the first kind are employed to construct the EqBFs and exponential functions are used as the weights in the integrals. The parameters of the solution are arranged so that all the integrals can be decomposed into much simpler 1D ones over a normalized intervals. This reduces the computational efforts significantly. Either the EqBFs or the results of the integration process may be stored for further use. The validity of the results is examined through an extended patch test. A set of physical sample problems with variable/non-variable material properties; as the potential flow over a cylinder, steady-state heat conduction problems in an anisotropic inhomogeneous functionally graded material, potential and stokes flow through a channel of finite width obstructed by periodic cylinders, and the bending of thin elastic plates having constant or variable thickness are solved to demonstrate the capabilities of the method. As a preliminary study, we show that the method may effectively be used in a domain decomposition approach. [Copyright &y& Elsevier]

Abstract: This paper presents a development of the direction splitting algorithm for flows in complex geometries proposed in [2] to the case of flows containing rigid particles. The main novelty of this method is that the grid is fit to the time-dependent domain at each time step which allows for an optimal spatial approximation of the Navier–Stokes equations. In general, this is a very hard task for multidimensional problems. However, it is significantly simplified if the momentum equations are discretized by a direction splitting scheme. Such a scheme requires only solving a set of one-dimensional problems, and the grid can be easily fit to the boundary in the direction of each of these problems. Here we use a MAC discretization stencil but the same idea can be applied to other discretizations. The equations of motion of each particle are discretized explicitly and the computed particle velocities are imposed as Dirichlet boundary conditions for the Navier–Stokes equations on the adapted grid. The pressure is extended within the particles in a fictitious domain fashion, but the divergence stencil is fit to the particle boundaries. In this paper, the direction splitting algorithm with boundary fitting is compared to pure fictitious domain methods. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re-used for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse-grid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method. [Copyright &y& Elsevier]

Abstract: Several studies have presented compact fourth order accurate finite difference approximation for the Helmholtz equation in two or three dimensions. Several of these formulae allow for the wave number k to be variable. Other papers have extended this further to include variable coefficients within the Laplacian which models non-homogeneous materials in electromagnetism. Later papers considered more accurate compact sixth order methods but these were restricted to constant k. In this paper we extend these compact sixth order schemes to variable k in both two and three dimensions. Results on 2D and 3D problems with known analytic solutions verify the sixth order accuracy. We demonstrate that for large wave numbers, the second order scheme cannot produce comparable results with reasonable grid sizes. [Copyright &y& Elsevier]

Abstract: This paper presents a new computational framework for the simulation of solid mechanics on general overlapping grids with adaptive mesh refinement (AMR). The approach, described here for time-dependent linear elasticity in two and three space dimensions, is motivated by considerations of accuracy, efficiency and flexibility. We consider two approaches for the numerical solution of the equations of linear elasticity on overlapping grids. In the first approach we solve the governing equations numerically as a second-order system (SOS) using a conservative finite-difference approximation. The second approach considers the equations written as a first-order system (FOS) and approximates them using a second-order characteristic-based (Godunov) finite-volume method. A principal aim of the paper is to present the first careful assessment of the accuracy and stability of these two representative schemes for the equations of linear elasticity on overlapping grids. This is done by first performing a stability analysis of analogous schemes for the first-order and second-order scalar wave equations on an overlapping grid. The analysis shows that non-dissipative approximations can have unstable modes with growth rates proportional to the inverse of the mesh spacing. This new result, which is relevant for the numerical solution of any type of wave propagation problem on overlapping grids, dictates the form of dissipation that is needed to stabilize the scheme. Numerical experiments show that the addition of the indicated form of dissipation and/or a separate filter step can be used to stabilize the SOS scheme. They also demonstrate that the upwinding inherent in the Godunov scheme, which provides dissipation of the appropriate form, stabilizes the FOS scheme. We then verify and compare the accuracy of the two schemes using the method of analytic solutions and using problems with known solutions. These latter problems provide useful benchmark solutions for time dependent elasticity. We also consider two problems in which exact solutions are not available, and use a posterior error estimates to assess the accuracy of the schemes. One of these two problems is additionally employed to demonstrate the use of dynamic AMR and its effectiveness for resolving elastic “shock” waves. Finally, results are presented that compare the computational performance of the two schemes. These demonstrate the speed and memory efficiency achieved by the use of structured overlapping grids and optimizations for Cartesian grids. [Copyright &y& Elsevier]

Abstract: Simulation of charged-particle beam devices using particle-in-cell methods in (r, z) cylindrical coordinates can be extremely efficient, incorporating a majority of the physics and yielding high resolution with minimal computational costs. However, accuracy requires adjustment of solution coefficients and particle/current weights with radius, especially near the cylindrical axis. Prior algorithms have been based on special models and approximations with no consistent method for computing coefficients in general cases. The Local Taylor Polynomial (LTP) technique presented in this paper provides a general framework for determination of coefficients and development of PDE solution algorithms. The LTP method employs a local (polynomial) solution of the continuum PDE to construct numerical algorithms. As a result, LTP systematically generates accurate coefficients and incorporates fields and sources on an equal basis. This paper focuses on the scalar Poisson PDE in cylindrical coordinates to illustrate the LTP technique, coefficient derivation, electric field calculation and handling of space charge effects. Application of LTP to non-conformal boundaries and non-uniform grids is presented. Although focused on cylindrical coordinates in this paper, the LTP technique has general applicability to other PDE’s and geometries, and is broadly applicable to problems in beam/field simulation. [Copyright &y& Elsevier]

Abstract: In this follow up paper to our previous study in Bayona et al. (2011) , we present a new technique to compute the solution of PDEs with the multiquadric based RBF finite difference method (RBF-FD) using an optimal node dependent variable value of the shape parameter. This optimal value is chosen so that, to leading order, the local approximation error of the RBF-FD formulas is zero. In our previous paper (Bayona et al., 2011) we considered the case of an optimal (constant) value of the shape parameter for all the nodes. Our new results show that, if one allows the shape parameter to be different at each grid point of the domain, one may obtain very significant accuracy improvements with a simple and inexpensive numerical technique. We analyze the same examples studied in Bayona et al. (2011) , both with structured and unstructured grids, and compare our new results with those obtained previously. We also find that, if there are a significant number of nodes for which no optimal value of the shape parameter exists, then the improvement in accuracy deteriorates significantly. In those cases, we use generalized multiquadrics as RBFs and choose the exponent of the multiquadric at each node to assure the existence of an optimal variable shape parameter. [Copyright &y& Elsevier]

Highlights • Hamilton–Jacobi equation. • Kernel based scheme. • Unconditionally stable. • High order accuracy. • Weighted essentially non-oscillatory methodology. Abstract In this paper, a class of high order numerical schemes is proposed for solving Hamilton–Jacobi (H–J) equations. This work is regarded as an extension of our previous work for nonlinear degenerate parabolic equations, see Christlieb et al. [14] , which relies on a special kernel-based formulation of the solutions and successive convolution. When applied to the H–J equations, the newly proposed scheme attains genuinely high order accuracy in both space and time, and more importantly, it is unconditionally stable, hence allowing for much larger time step evolution compared with other explicit schemes and saving computational cost. A high order weighted essentially non-oscillatory methodology and a novel nonlinear filter are further incorporated to capture the correct viscosity solution. Furthermore, by coupling the recently proposed inverse Lax–Wendroff boundary treatment technique, this method is very flexible in handing complex geometry as well as general boundary conditions. We perform numerical experiments on a collection of numerical examples, including H–J equations with linear, nonlinear, convex or non-convex Hamiltonians. The efficacy and efficiency of the proposed scheme in approximating the viscosity solution of general H–J equations is verified. [ABSTRACT FROM AUTHOR]

Abstract We propose a new tool for the study of the parabolic p -curl problem modelling the magnetic field in a high-temperature superconductor (HTS), a problem involving an operator ∇ × (| ∇ × H | p − 2 ∇ × H) analogous to the p -Laplacian. This so-called nonlinear p -curl problem is computationally expensive in three space dimensions because sharp fronts in the magnetic field develop near the surface of HTS. The proposed technique is a relaxation model of the p -curl problem which is shown to provide a consistent and stable approximation according to first, an inner/outer layer analysis and secondly, according to a time scale expansion. The relaxation model leads to monotone approximations of the front because of the presence of anisotropic diffusion. Most importantly, it can be used to develop new schemes that inherit the properties of the relaxation model. In order to demonstrate the relevance of the relaxation model, this paper applies the approximation to the construction of a second-order nonlinear finite-difference time-domain (FDTD) method, similar to Yee's scheme. It is shown that the natural discretization of the p -curl inspired by Yee's scheme leads to an unstable scheme, while a discretization based on the relaxation model is stable. We propose two numerical discretizations that are respectively first and second order in time, but both second-order in space. The linear and nonlinear stability of the schemes is studied and it is demonstrated that the schemes are both discrete divergence free. We verify the numerical schemes using manufactured solutions and Mayergoyz's moving-front solution in both one and two space dimensions, as well as on a magnetization of a typical HTS. [ABSTRACT FROM AUTHOR]

Highlights • Accurate and efficient numerical simulation of granular flow. • Two regularized models of the μ (I) -rheology. • Application of the PFEM to frictional material simulation. • Validation against 3D experimental tests. Abstract This paper proposes two regularized models of the μ (I) -rheology and shows their application to the numerical simulation of 3D dense granular flows. The proposed regularizations are inspired by the Papanastasiou and Bercovier–Engleman methods, typically used to approximate the Bingham law. The key idea is to keep limited the value of the apparent viscosity for low shear rates without introducing a fixed cutoff. The proposed techniques are introduced into the Particle Finite Element Method (PFEM) framework to deal with the large deformations expected in free-surface granular flows. After showing the numerical drawbacks associated to the standard μ (I) -rheology, the two regularization strategies are derived and discussed. The regularized μ (I) -rheology is then applied to the simulation of the collapse of 2D and 3D granular columns. The numerical results show that the regularization techniques improve substantially the conditioning of the linear system without affecting the solution accuracy. A good agreement with the experimental tests and other numerical methods is obtained in all the analyzed problems. [ABSTRACT FROM AUTHOR]

Abstract The Harten–Lax–van Leer with contact (HLLC) approximate Riemann solver is known to be plagued by various forms of numerical shock instability. In this paper, we propose a new framework for developing shock stable versions of the HLLC scheme for the Euler system of equations without compromising on its contact and shear preserving ability. The proposed framework, termed as S elective W ave M odification (SWM), identifies and enhances the magnitude of the inherent HLL-type diffusive component within the HLLC scheme in the vicinity of a normal shock front. This is achieved by suitably increasing the magnitudes of the nonlinear wave speed estimates appearing in them through a dissipation parameter ϵ. A linear perturbation analysis is performed to gauge the effectiveness of the proposed framework in damping unwanted perturbations in physical quantities and is also used to estimate a von-Neumann-type stability bound on the CFL number associated with its use. Two distinct methods to estimate ϵ , which results in HLLC-SWM-E (Eigenvalue based) and HLLC-SWM-P (Pressure based) schemes, are proposed and compared. Further, a matrix based stability analysis is used to show that the proposed schemes can be configured to remain shock stable over a wide range of free stream Mach numbers. Numerical results confirm that the proposed schemes are capable of computing shock stable solutions for a wide variety of problems. On viscous flows, the HLLC-SWM-P variant is found to be the most accurate. Performance of these schemes suggests that it is possible to construct shock stable upwind schemes that simultaneously retains accuracy on the linearly degenerate wavefields. Highlights • A simple strategy to cure shock instability in the HLLC Riemann solver proposed. • Cure achieved by enhancing the inherent HLL-type diffusive component. • Two shock stable variants namely HLLC-SWM-E and HLLC-SWM-P are developed. • Robustness and accuracy of the scheme are demonstrated on several numerical examples. [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on the question of how unconditional stability can be achieved via multistep ImEx schemes, in practice problems where both the implicit and explicit terms are allowed to be stiff. For a class of new ImEx multistep schemes that involve a free parameter, strategies are presented on how to choose the ImEx splitting and the time stepping parameter, so that unconditional stability is achieved under the smallest approximation errors. These strategies are based on recently developed stability concepts, which also provide novel insights into the limitations of existing semi-implicit backward differentiation formulas (SBDF). For instance, the new strategies enable higher order time stepping that is not otherwise possible with SBDF. With specific applications in nonlinear diffusion problems and incompressible channel flows, it is demonstrated how the unconditional stability property can be leveraged to efficiently solve stiff nonlinear or nonlocal problems without the need to solve nonlinear or nonlocal problems implicitly. Highlights • Unconditional stability for equations with both implicit and explicit stiff terms. • Explanations of stability limitations in semi-implicit backward differentiation. • Strategies for the simultaneous choice of splitting and time stepping coefficients. • Examples avoiding nonlinear implicit solves in stiff nonlinear diffusion equations. [ABSTRACT FROM AUTHOR]

Abstract In this paper we analyze a multilevel quasidiffusion (QD) method for solving time-dependent multigroup nonlinear radiative transfer problems which describe interaction of photons with matter. The multilevel method is formulated by means of the high-order radiative transfer equation and a set of low-order moment equations. The fully implicit scheme is used to discretize equations in time. The stability analysis is applied to the method in semi-continuous and discretized forms. To perform Fourier analysis, the system of equations of the multilevel method is linearized about an equilibrium solution. The effects of discretization with respect to different independent variables are studied. The multilevel method is shown to be stable and fast converging. We also consider a version of the method in which time evolution in the radiative transfer equation is treated by means of the α -approximation. The Fleck–Cummings test problem is used to demonstrate performance of the multilevel QD method and study its iterative stability. Highlights • Fast multilevel iterative method for solving time-dependent nonlinear radiative transfer problems. • Fourier analysis of the multilevel quasidiffusion method in semi-continuous and discretized forms. • Performance and stability of the multilevel method are analyzed on a model multigroup thermal radiative transfer problem. [ABSTRACT FROM AUTHOR]

Abstract: The study illustrated in this paper completes the topics initially investigated in Ref. , the aim being here to analyze the role of the integral-based Finite Volume (FV) discretizations in Large Eddy Simulations that exploit the implicit filtering approach. Specifically, a theoretical study on the effective shape and length of three-dimensional filters induced by some FV-based flux reconstructions is the object of this paper. For any integral-based flux reconstruction, one gets always an approximation of the top-hat filter kernel. This is not the case of the filters induced by the differential-based Finite Difference operators, such as those reported and analyzed in Refs. . Considering the sub-filter resolution parameter Q =Δ eff /h, being Δ eff the effective filter width and h the computational grid size, allows us discerning the effective measure of the approximate built-in top-hat filter. The induced shape and width is analyzed by means of a modified wavenumber-like analysis that is developed in the 3D Fourier space. Several evaluation criteria applied on different schemes are considered and the differences in terms of either velocity or flux interpolations on staggered or non-staggered grids are analyzed. Conclusions are reported that, depending on the using of either the integral or the differential form of the filtered equations, the induced numerical filter is or is not a congruent approximation of the exact top-hat transfer function for some value Q. The need of a suitable estimation of the sub-filter parameter Q is assessed from several real LES computations, that make use of the new integral-based version of the eddy-viscosity dynamic modeling presented in Ref. . In fact, it is shown that the test-filtering length has to be carefully chosen as a function of the FV-based induced filter. [Copyright &y& Elsevier]

Abstract: The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L 2-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a semi-Lagrangian finite difference formulation for approximating conservative form of advection equations with general variable coefficients. Compared with the traditional semi-Lagrangian finite difference schemes , which approximate the advective form of the equation via direct characteristics tracing, the scheme proposed in this paper approximates the conservative form of the equation. This essential difference makes the proposed scheme naturally conservative for equations with general variable coefficients. The proposed conservative semi-Lagrangian finite difference framework is coupled with high order essentially non-oscillatory (ENO) or weighted ENO (WENO) reconstructions to achieve high order accuracy in smooth parts of the solution and to capture sharp interfaces without introducing spurious oscillations. The scheme is extended to high dimensional problems by Strang splitting. The performance of the proposed schemes is demonstrated by linear advection, rigid body rotation, swirling deformation, and two dimensional incompressible flow simulation in the vorticity stream-function formulation. As the information is propagating along characteristics, the proposed scheme does not have the CFL time step restriction of the Eulerian method, allowing for a more efficient numerical realization for many application problems. [ABSTRACT FROM AUTHOR]

Abstract In the interpolation method, in some cases, one often has a number of data points and its derivatives, which are obtained by sampling or experimentation. In this case, the problem of finding an approximating function passing through these points and coinciding with given values of its derivatives at these points is generally known as "Hermite interpolation". The Hermite interpolation is mostly a method of interpolating data points as a polynomial function that is faced with some challenges in high dimensions and on irregular domains. Radial basis functions take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. So, they can be applied as a suitable tool to high dimensional Hermite interpolation problem on irregular domains. In many applied systems, commonly available derivatives information is presented using fractional order derivatives instead of integer ones. For this purpose, in this paper, we assume that the values of an unknown function and its fractional derivatives at some distinct points are presented. Therefore, we intend to apply a new approach, which we call it as "fractional Hermite interpolation" with radial basis functions in high dimensions. Optimal recovery conditions for the fractional Hermite interpolant are investigated. Then, the existence and uniqueness of the solution in this type of generalized interpolation are proved. In order to increase the accuracy and stability of the method, Hilbert Schmidt's theory has also been used. The main advantages of the used method are its simplicity and efficiency in high dimensions, and over irregular domains. Finally, numerical results in one, two and three dimensions and a real-world problem are presented to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract While high order methods became very popular as they allow to perform very accurate solutions with low computational time and memory cost, there is a lack of tools to visualize and post-treat the solutions given by these methods. Originally, visualization softwares were developed to post-process results from methods such that finite differences or usual finite elements and therefore process linear primitives. In this paper, we present a methodology to visualize results of high order methods. Our approach is based on the construction of an optimized affine approximation of the high order solution which can therefore be handled by any visualization software. A representation mesh is constructed and the process is guided by an a posteriori estimate which control the error between the numerical solution and its representation pointwise. This point by point control is crucial as under their picture form, data correspond to values mapped on elements where anyone can pick up a pointwise information. A strategy is established to ensure that discontinuities are well represented. These discontinuities come either from the physical problem (material change) or the numerical method (Discontinuous Galerkin method) and are pictured accurately. Several numerical examples are presented to demonstrate the potential of the method. Highlights • Objectives to faithfully represent high order solutions are defined. • An algorithm to obtain the representation is given. • Under assumptions, the representation is proved to satisfy the objectives. • Specific developments for approximation spaces based on curved elements are led. • Relevancy and benefits are outlined with several numerical examples. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider numerical methods for solving the two-dimensional Vlasov–Poisson equation in the finite Larmor radius approximation regime. The model describes the behaviour of charged particles under a strong external magnetic field and the finite Larmor radius scaling. We discretise the equation under Particle-in-Cell method, where the characteristics equations are highly oscillatory system in the limit regime. We apply popular numerical integrators including splitting methods, multi-revolution composition methods, two-scale formulation method and limit solver to integrate the characteristics. We then highlight the strengths and drawbacks of each method. Finally, numerical experiments are presented, and comparisons on the accuracy, efficiency and long-time behaviour of the methods are made, pointing to the method with the best performance for this problem. [ABSTRACT FROM AUTHOR]

Abstract Quadrature by Expansion (QBX) is a quadrature method for approximating the value of the singular integrals encountered in the evaluation of layer potentials. It exploits the smoothness of the layer potential by forming locally-valid expansions which are then evaluated to compute the near or on-surface value of the potential. Recent work towards coupling of a Fast Multipole Method (FMM) to QBX yielded a first step towards the rapid evaluation of such integrals (and the solution of related integral equations), albeit with only empirically understood error behavior. In this paper, we improve upon this approach with a modified algorithm for which we give a comprehensive analysis of error and cost in the case of the Laplace equation in two dimensions. For the same levels of (user-specified) accuracy, the new algorithm empirically has cost-per-accuracy comparable to prior approaches. We provide experimental results to demonstrate scalability and numerical accuracy. Highlights • A new fast multipole algorithm for Quadrature by Expansion (QBX). • Comprehensive error and cost analysis for the case of the 2D Laplace kernel. • Empirical results match modeled error and scaling behavior. [ABSTRACT FROM AUTHOR]

Abstract In this paper, an advanced numerical technique is proposed for a low-frequency electromagnetic field simulation. The finite difference time domain (FDTD) method is one of the best methods for analyzing electromagnetic field problems, which require high levels of precision, such as a heterogeneous whole-body voxel human model. However, directly using the standard FDTD method in the quasi-static frequency band requires an extra-large number of iterations. We overcome this drawback using the hybrid of surface equivalence theorem and quasi-static FDTD (QS-FDTD) method. The surface equivalence theorem is used to consider arbitrary shape of source and the QS-FDTD is applied to obtain electromagnetic response quickly in the low-frequency region. The results of the proposed method are in good agreement with those from a commercial electromagnetic simulator. The proposed method is valid where the quasi-static approximation is satisfied. [ABSTRACT FROM AUTHOR]

Finite Element Methods (FEM) are becoming increasingly popular in modeling seismic wave propagation. These methods provide higher order accuracy, geometrical flexibility and adaptive gridding capabilities that are not easy to incorporate in traditional finite difference methods employed for generation of synthetic seismograms. Moreover, several studies have shown that Discontinuous Galerkin FEM (DGM) is a promising approach for modeling wave propagation in fractured media. Here we propose an Enriched Galerkin FEM (EGM) for elastic wave propagation. EGM uses the same bilinear form as DGM and the continuous Galerkin finite element spaces enriched by discontinuous piecewise constants or bilinear functions. EGM satisfies local equilibrium while reducing the degrees of freedom in DGM formulations. In this paper, we consider elastic wave propagation and derive optimal a priori error estimates for DGM and EGM. We present numerical examples in two spatial dimensions that confirm these theoretical results. In addition, we provide numerical comparisons with the Spectral element method. In previous work, DGM has been shown to be effective in modeling elastic wave propagation in fractured media using the linear slip model. We now extend these results to EGM with reduced computational costs over DGM. [ABSTRACT FROM AUTHOR]

We are interested in the approximation of a steady hyperbolic problem. In some cases, the solution can satisfy an additional conservation relation, at least when it is smooth. This is the case of an entropy. In this paper, we show, starting from the discretization of the original PDE, how to construct a scheme that is consistent with the original PDE and the additional conservation relation. Since one interesting example is given by the systems endowed by an entropy, we provide one explicit solution, and show that the accuracy of the new scheme is at most degraded by one order. In the case of a discontinuous Galerkin scheme and a Residual distribution scheme, we show how not to degrade the accuracy. This improves the recent results obtained in [1–4] in the sense that no particular constraints are set on quadrature formula and that a priori maximum accuracy can still be achieved. We study the behavior of the method on a non linear scalar problem. However, the method is not restricted to scalar problems. [ABSTRACT FROM AUTHOR]

We develop new approximations of the Mori–Zwanzig equation based on operator series expansions of the orthogonal dynamics propagator. In particular, we study the Faber polynomial series, which yields asymptotically optimal approximations converging at least R -superlinearly with the polynomial order. Numerical applications are presented for random wave propagation and harmonic chains of oscillators interacting on the Bethe lattice and on graphs with arbitrary topology. Most of the analysis and numerical results presented in this paper are for linear dynamical systems evolving from random initial states. The extension to nonlinear systems is discussed as well the challenges in the implementation of the proposed approximation methods. [ABSTRACT FROM AUTHOR]

For a large class of orthogonal basis functions, there has been a recent identification of expansion methods for computing accurate, stable approximations of a quantity of interest. This paper presents, within the context of uncertainty quantification, a practical implementation using basis adaptation, and coherence motivated sampling, which under assumptions has satisfying guarantees. This implementation is referred to as Basis Adaptive Sample Efficient Polynomial Chaos (BASE-PC). A key component of this is the use of anisotropic polynomial order which admits evolving global bases for approximation in an efficient manner, leading to consistently stable approximation for a practical class of smooth functionals. This fully adaptive, non-intrusive method, requires no a priori information of the solution, and has satisfying theoretical guarantees of recovery. A key contribution to stability is the use of a presented correction sampling for coherence-optimal sampling in order to improve stability and accuracy within the adaptive basis scheme. Theoretically, the method may dramatically reduce the impact of dimensionality in function approximation, and numerically the method is demonstrated to perform well on problems with dimension up to 1000. [ABSTRACT FROM AUTHOR]

The presence of a high-dimensional stochastic input domain with discontinuities poses major computational challenges in analyzing and quantifying the effects of the uncertainties in a physical system. In this paper, we propose a stochastic collocation method with adaptive mesh refinement (SCAMR) to deal with high dimensional stochastic systems with discontinuities. Specifically, the proposed approach uses generalized polynomial chaos (gPC) expansion with Legendre polynomial basis and solves for the gPC coefficients using the least squares method. It also implements an adaptive mesh (element) refinement strategy which checks for abrupt variations in the output based on a low-order gPC approximation error to track discontinuities or non-smoothness. In addition, the proposed method involves a criterion for checking possible dimensionality reduction and consequently, the decomposition of the original high-dimensional problem to a number of lower-dimensional subproblems. Specifically, this criterion checks all the existing interactions between input parameters of a specific problem based on the high-dimensional model representation (HDMR) method, and therefore automatically provides the subproblems which only involve interacting input parameters. The efficiency of the approach is demonstrated using examples of both smooth and non-smooth problems with number of input parameters up to 500, and the approach is compared against other existing algorithms. [ABSTRACT FROM AUTHOR]

The aim of this paper is to compute the mean field and variance of solutions to three-dimensional Maxwell's equations with random interfaces via shape calculus and pivoted low-rank approximation. Based on the perturbation theory and shape calculus, we characterize the statistical moments of solutions to Maxwell's equations with random interfaces in terms of the perturbation magnitude via the first order shape-Taylor expansion. In order to capture oscillations with high resolution close to the interface, an adaptive finite element method using Nédélec's third order edge elements of the first kind is employed to solve the deterministic Maxwell's equations with the mean interface to approximate the expectation of solutions. For the second moment computation, an efficient low-rank approximation of the pivoted Cholesky decomposition is proposed to compute the two-point correlation function to approximate the variance of solutions. Numerical experiments are presented to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper serves to treat boundary conditions numerically with high order accuracy in order to suit the two-stage fourth-order finite volume schemes for hyperbolic problems developed in J. Li and Z. Du (2016) [17] . As such, it is significant when capturing small scale structures near physical boundaries. Different from previous contributions in literature, the current approach constructs a fourth order accurate approximation to boundary conditions by only using the Jacobian matrix of the flux function (characteristic information) instead of its successive differentiation of governing equations leading to tensors of high ranks in the inverse Lax–Wendroff method. Technically, data in several ghost cells are constructed with interpolation so that the interior scheme can be implemented over boundary cells, and theoretical boundary condition has to be modified properly at intermediate stages so as to make the two-stage scheme over boundary cells fully consistent with that over interior cells. This is nonintuitive and highlights the fact that theoretical boundary conditions are only prescribed for continuous partial differential equations (PDEs), while they must be approximated in a consistent way (even though they could be exactly valued) when the PDEs are discretized . Several numerical examples are provided to illustrate the performance of the current approach when dealing with general boundary conditions. [ABSTRACT FROM AUTHOR]

The asymptotic, physical optics (PO) approximation is applicable to electromagnetic scattering analysis of electrically large, conducting objects, which is important for various applications, such as radar cross-section (RCS) calculations. It relates the incident magnetic field to the induced surface current at points on a scatterer's surface with line-of-sight visibility to the source. Here, the focus is on mesh-based PO, using a triangle element mesh with standard Rao–Wilton–Glisson basis functions to represent the surface current density. The multiple-reflection PO (MRPO) method involves applying the PO approximation to the field radiated by the present current solution towards the scatterer itself, to account for successive internal reflections. The visibility status between every pair of basis functions is required, taking into account all geometry. In a conventional implementation, internal reflected field calculation and internal shadowing determination runtimes both scale as O ( N 2 ) ; N denotes the number of mesh elements. In this paper an accelerated version of the mesh-based MRPO method is presented: fast MRPO (FMRPO). The multi-level, fast multipole method (MLFMM) is used to accelerate internal reflected field calculation. The key aspect to achieving acceleration for general geometries, is to incorporate internal shadowing into the MLFMM interaction tree. This requires an alteration to the inter-group interaction criterion of the MLFMM, as well as efficient evaluation of inter-group shadowing status flags, to preserve the beneficial cost-scaling property of the MLFMM. Algorithmic parameters are introduced, which control the accuracy of internal shadowing determination. The FMRPO runtime scales as O(NlogN), in the high-frequency limit. Numerical results are presented, demonstrating accuracy and efficiency for general scattering objects. [ABSTRACT FROM AUTHOR]

Reconstructing the interface within a cell, based on volume fraction and normal direction, is a key part of multiphase flow solvers which make use of piecewise linear interface calculation (PLIC) such as the Coupled Level Set Volume of Fluid (CLSVOF) method. In this paper, we present an iterative method for interface reconstruction (IR) in general convex cells based on tetrahedral decomposition. By splitting the cell into tetrahedra prior to IR the volume of the truncated polyhedron can be calculated much more rapidly than using existing clipping and capping methods. In addition the root finding algorithm is designed to take advantage of the nature of the relationship between volume fraction and interface position by using a combination of Newton's and Muller's methods. In stand-alone tests of the IR algorithm on single cells with up to 20 vertices the proposed method was found to be 2 times faster than an implementation of an existing analytical method, while being easy to implement. It was also found to be 3.4–11.8 times faster than existing iterative methods using clipping and capping and combined with Brent's root finding method. Tests were then carried out of the IR method as part of a CLSVOF solver. For a sphere deformed by a prescribed velocity field the proposed method was found to be up to 33% faster than existing iterative methods. For simulations including the solution of the velocity field the maximum speed up was found to be approximately 52% for a case where 12% of cells lie on the interface. Analysis of the full simulation CPU time budget also indicates that while the proposed method has produced a considerable speed-up, further gains due to increasing the efficiency of the IR method are likely to be small as the IR step now represents only a small proportion of the run time. [ABSTRACT FROM AUTHOR]

In this paper we develop the a posteriori error analysis of an augmented mixed-primal finite element method for the 2D and 3D versions of a stationary flow and transport coupled system, typically encountered in sedimentation–consolidation processes. The governing equations consist in the Brinkman problem with concentration-dependent viscosity, written in terms of Cauchy pseudo-stresses and bulk velocity of the mixture; coupled with a nonlinear advection – nonlinear diffusion equation describing the transport of the solids volume fraction. We derive two efficient and reliable residual-based a posteriori error estimators for a finite element scheme using Raviart–Thomas spaces of order k for the stress approximation, and continuous piecewise polynomials of degree ≤ k + 1 for both velocity and concentration. For the first estimator we make use of suitable ellipticity and inf–sup conditions together with a Helmholtz decomposition and the local approximation properties of the Clément interpolant and Raviart–Thomas operator to show its reliability, whereas the efficiency follows from inverse inequalities and localisation arguments based on triangle-bubble and edge-bubble functions. Next, we analyse an alternative error estimator, whose reliability can be proved without resorting to Helmholtz decompositions. Finally, we provide some numerical results confirming the reliability and efficiency of the estimators and illustrating the good performance of the associated adaptive algorithm for the augmented mixed-primal finite element method. [ABSTRACT FROM AUTHOR]

Abstract: The kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically-conducting liquid. One of its main uses is as a proof-of-concept tool to test hypotheses about self-exciting dynamo action. Indeed, it provided the very first quantitative evidence for the possibility of the geodynamo. Despite its utility, due to the requirement of resolving fine structures, historically, numerical work has proven difficult and reported solutions were often plagued by poor convergence. In this paper, we demonstrate the numerical superiority of a Galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere. After adopting a poloidal–toroidal decomposition and expanding in spherical harmonics, we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials. Each basis function is constructed from a terse sum of one-sided Jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior, but is everywhere infinitely differentiable, including at the origin. This Galerkin method exhibits more rapid convergence, for a given problem size, than any other scheme hitherto reported, as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature. In the axisymmetric flows we consider in this paper, at a magnetic Reynolds number of O(100), a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions; alternatively, 4 significant figures requires 20 radial functions. The terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics. We exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of Bachtiar et al. , thereby verifying a counter-example to the Zel’dovich anti-dynamo theorem in a spherical geometry. [Copyright &y& Elsevier]

Abstract: A key idea in finite difference weighted essentially non-oscillatory (WENO) schemes is a combination of lower order fluxes to obtain a higher order approximation. The choice of the weight to each candidate stencil, which is a nonlinear function of the grid values, is crucial to the success of WENO schemes. For the system case, WENO schemes are based on local characteristic decompositions and flux splitting to avoid spurious oscillation. But the cost of computation of nonlinear weights and local characteristic decompositions is very high. In this paper, we investigate hybrid schemes of WENO schemes with high order up-wind linear schemes using different discontinuity indicators and explore the possibility in avoiding the local characteristic decompositions and the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong shocks. The idea is to identify discontinuity by an discontinuity indicator, then reconstruct numerical flux by WENO approximation in discontinuous regions and up-wind linear approximation in smooth regions. These indicators are mainly based on the troubled-cell indicators for discontinuous Galerkin (DG) method which are listed in the paper by Qiu and Shu (J. Qiu, C.-W. Shu, A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin methods using weighted essentially non-oscillatory limiters, SIAM Journal of Scientific Computing 27 (2005) 995–1013). The emphasis of the paper is on comparison of the performance of hybrid scheme using different indicators, with an objective of obtaining efficient and reliable indicators to obtain better performance of hybrid scheme to save computational cost. Detail numerical studies in one- and two-dimensional cases are performed, addressing the issues of efficiency (less CPU time and more accurate numerical solution), non-oscillatory property. [ABSTRACT FROM AUTHOR]

Abstract: With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general anisotropic media, we investigate the accuracy aspects of local ABCs designed for the scalar anisotropic wave equation in the frequency domain (time harmonic case). The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL). PMDL is a simple variant of perfectly matched layers (PML) and is equivalent to rational approximation-based local ABCs. Specifically, we derive a sufficient condition for PMDL to accurately absorb wave modes with outgoing group velocities and this condition turns out to be a simple bound on the PMDL parameters. The reflection coefficient derived in this paper clearly reveals that the PMDL absorption is based on group velocities, and not phase velocities, and hence a PMDL can be designed to correctly identify and accurately absorb all outgoing wave modes (even those with opposing signs of phase and group velocities). The validity of the sufficient condition is demonstrated through a series of frequency domain simulations. In part 2 of this paper [S. Savadatti, M.N. Guddati, Absorbing boundary conditions for scalar waves in anisotropic media. Part 2: Time-dependent modeling, J. Comput. Phys. (2010), doi:10.1016/j.jcp.2010.05.017], the accuracy condition presented here is shown to govern both the well-posedness and accuracy aspects of PMDL designed for transient (time-dependent) modeling of scalar waves in anisotropic media. [ABSTRACT FROM AUTHOR]

Abstract: With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general anisotropic media, we investigate the well-posedness and accuracy aspects of local ABCs designed for the transient modeling of the scalar anisotropic wave equation. The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL), a simple variant of perfectly matched layers (PML) that is also equivalent to rational approximation based ABCs. Specifically, we derive the necessary and sufficient condition for the well-posedness of the initial boundary value problem (IBVP) obtained by coupling an interior and a PMDL ABC. The derivation of the reflection coefficient presented in a companion paper (S. Savadatti, M.N. Guddati, J. Comput. Phys., 2010, doi:10.1016/j.jcp.2010.05.018) has shown that PMDL can correctly identify and accurately absorb outgoing waves with opposing signs of group and phase velocities provided the PMDL layer lengths satisfy a certain bound. Utilizing the well-posedness theory developed by Kreiss for general hyperbolic IBVPs, and the well-posedness conditions for ABCs derived by Trefethen and Halpern for isotropic acoustics, we show that this bound on layer lengths also ensures well-posedness. The time discretized form of PMDL is also shown to be theoretically stable and some instability related to finite precision arithmetic is discussed. [Copyright &y& Elsevier]

Abstract: The recently developed Flexible Local Approximation MEthod (FLAME) produces accurate difference schemes by replacing the usual Taylor expansion with Trefftz functions – local solutions of the underlying differential equation. This paper advances and casts in a general form a significant modification of FLAME proposed recently by Pinheiro and Webb: a least-squares fit instead of the exact match of the approximate solution at the stencil nodes. As a consequence of that, FLAME schemes can now be generated on irregular stencils with the number of nodes substantially greater than the number of approximating functions. The accuracy of the method is preserved but its robustness is improved. For demonstration, the paper presents a number of numerical examples in 2D and 3D: electrostatic (magnetostatic) particle interactions, scattering of electromagnetic (acoustic) waves, and wave propagation in a photonic crystal. The examples explore the role of the grid and stencil size, of the number of approximating functions, and of the irregularity of the stencils. [Copyright &y& Elsevier]

Abstract: We develop in this paper a model for the numerical simulation of free surface debris- and mudflows, described as a mixture of two viscous phases. A Lagrangian approach with finite element approximation and regular remeshing is adopted in this paper. The proposed model solves the equations of motion in both phases, coupled with conservation of volume of the mixture, implicitly; volume fractions are computed explicitly at the end of each iteration. A one-step trapezoidal scheme is used for time integration. The resulting model combines simplicity, robustness and versatility and provides a novel and unified algorithmic framework for a large spectrum of applications. It is also easily extensible to three spatial dimensions, more complex constitutive models and coupling with geomechanical models. Application to sedimentation and to impact on an obstacle shows the effectiveness of the approach. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the numerical approximation of the isothermal Euler equations for charged particles subject to the Lorentz force (the ‘Euler–Lorentz’ system). When the magnetic field is large, or equivalently, when the parameter representing the non-dimensional ion cyclotron frequency tends to zero, the so-called drift-fluid (or gyro-fluid) approximation is obtained. In this limit, the parallel motion relative to the magnetic field direction splits from perpendicular motion and is given implicitly by the constraint of zero total force along the magnetic field lines. In this paper, we provide a well-posed elliptic equation for the parallel velocity which in turn allows us to construct an Asymptotic-Preserving (AP) scheme for the Euler–Lorentz system. This scheme gives rise to both a consistent approximation of the Euler–Lorentz model when is finite and a consistent approximation of the drift limit when . Above all, it does not require any constraint on the space and time-steps related to the small value of . Numerical results are presented, which confirm the AP character of the scheme and its Asymptotic Stability. [Copyright &y& Elsevier]

Abstract: A set of optimised boundary closure schemes is presented for use with compact central finite difference schemes in computational aeroacoustics (CAA) involving non-trivial boundaries. The boundary schemes are given in a form of non-central compact finite differences. They maintain fourth-order accuracy, a pentadiagonal matrix system and seven-point stencil which the main interior scheme employs. This paper introduces a new strategy to optimise the boundary schemes in the spectral domain and achieve the best resolution characteristics given a strict tolerance for the dispersion and dissipation errors. The boundary schemes are derived from sophisticated extrapolation of solutions outside the domain. The extrapolation functions are devised by combining polynomials and trigonometric series which contain extra control variables used to optimise the resolution characteristics. The differencing coefficients of the boundary schemes are determined in association with the existing coefficients of the interior scheme which is also optimised through an improved procedure in this paper. The accuracy of the proposed schemes is demonstrated by their application to CAA benchmark problems. [Copyright &y& Elsevier]

Models for reacting flow are typically based on advection–diffusion–reaction (A–D–R) partial differential equations. Many practical cases correspond to situations where the relevant time scales associated with each of the three sub-processes can be widely different, leading to disparate time-step requirements for robust and accurate time-integration. In particular, interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection. The numerical strategy introduced in this paper is a general procedure to account for this time-scale disparity. The proposed methods are high-order multi-implicit generalizations of spectral deferred correction methods (MISDC methods), constructed for the temporal integration of A–D–R equations. Spectral deferred correction methods compute a high-order approximation to the solution of a differential equation by using a simple, low-order numerical method to solve a series of correction equations, each of which increases the order of accuracy of the approximation. The key feature of MISDC methods is their flexibility in handling several sub-processes implicitly but independently, while avoiding the splitting errors present in traditional operator-splitting methods and also allowing for different time steps for each process. The stability, accuracy, and efficiency of MISDC methods are first analyzed using a linear model problem and the results are compared to semi-implicit spectral deferred correction methods. Furthermore, numerical tests on simplified reacting flows demonstrate the expected convergence rates for MISDC methods of orders three, four, and five. The gain in efficiency by independently controlling the sub-process time steps is illustrated for nonlinear problems, where reaction and diffusion are much stiffer than advection. Although the paper focuses on this specific time-scales ordering, the generalization to any ordering combination is straightforward. [Copyright &y& Elsevier]

The advection equation is the basis for mathematical models of continuum mechanics. In the approximate solution of nonstationary problems it is necessary to inherit main properties of the conservatism and monotonicity of the solution. In this paper, the advection equation is written in the symmetric form, where the advection operator is the half-sum of advection operators in conservative (divergent) and non-conservative (characteristic) forms. The advection operator is skew-symmetric. Standard finite element approximations in space are used. The standard explicit two-level scheme for the advection equation is absolutely unstable. New conditionally stable regularized schemes are constructed, on the basis of the general theory of stability (well-posedness) of operator-difference schemes, the stability conditions of the explicit Lax–Wendroff scheme are established. Unconditionally stable and conservative schemes are implicit schemes of the second (Crank–Nicolson scheme) and fourth order. The conditionally stable implicit Lax–Wendroff scheme is constructed. The accuracy of the investigated explicit and implicit two-level schemes for an approximate solution of the advection equation is illustrated by the numerical results of a model two-dimensional problem. [ABSTRACT FROM AUTHOR]

This paper presents two discrete maximum principles (DMP) for the numerical solution of second order elliptic equations arising from the weak Galerkin finite element method. The results are established by assuming an h -acute angle condition for the underlying finite element triangulations. The mathematical theory is based on the well-known De Giorgi technique adapted in the finite element context. Some numerical results are reported to validate the theory of DMP. [ABSTRACT FROM AUTHOR]

The Bloch mode synthesis (BMS) model-reduction method adapts component mode synthesis techniques to unit-cell problems in order to obtain a reduced-order model that quickly produces band-structure frequencies for any wave vector, or vice versa. Fundamental to BMS is a partitioning of the real-space model into interior and boundary components, and subsequent reduction of the interior via truncated normal mode expansion. In this paper, two enhancements are presented for the BMS method that reduce both computation time and error in band-structure calculations. The first enhancement improves the accuracy of the interior reduction by approximating the participation of the residual modes rather than simply truncating them. The original formulation of BMS includes a modal reduction of the boundary that must be recomputed for every wave vector. This limits computational benefits and prevents the reduced-order model from being useful for the inverse band-structure problem (i.e., the k ( ω ) calculation). The second enhancement is a local boundary reduction that is independent of wave vector and thus does not suffer from the aforementioned limitations. [ABSTRACT FROM AUTHOR]

A new algorithm for numerical evaluation of spherical Bessel transform is proposed in this paper. In this method, the spherical Bessel function is approximately represented as an exponential sum with complex parameters. This is obtained by expressing an integral representation of spherical Bessel function in complex plane, and discretizing contour integrals along steepest descent paths and a contour path parallel to real axis using numerical quadrature rule with the double-exponential transformation. The number of terms in the expression is reduced using the modified balanced truncation method. The residual part of integrand is also expanded by exponential functions using Prony-like method. The spherical Bessel transform can be evaluated analytically on arbitrary points in half-open interval. [ABSTRACT FROM AUTHOR]

This paper describes a high order accurate method to calculate integrals over curved surfaces with boundaries. Given data locations that are arbitrarily distributed over the surface, together with some functional description of the surface and its boundary, the algorithm produces matching quadrature weights. This extends on the authors' earlier methods for integrating over the surface of a sphere and over arbitrarily shaped smooth closed surfaces by also considering domain boundaries. The core approach consists again of combining RBF-FD (radial basis function-generated finite difference) approximations for curved surface triangles, which together make up the full surface. The provided examples include both curved and flat domains. In the highly special case of equi-spaced nodes over a regular interval in 1-D, the method provides a new opportunity for improving on the classical Gregory enhancements of the trapezoidal rule. [ABSTRACT FROM AUTHOR]