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• A novel positivity-preserving algorithm for implicit NS solver. • A residual correction to compute the correction factor. • A flux correction to enforce the positivity of the solution conservatively. • Positivity-preserving combined with implicit iterations. • Numerical experiments to verify the positivity-preserving capability. This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes that solve compressible Euler and Navier-Stokes equations to ensure the positivity of density and internal energy (or pressure). Previous positivity-preserving algorithms are mainly based on the slope limiting or flux limiting technique, which rely on the existence of low-order positivity-preserving schemes. This dependency poses serious restrictions on extending these algorithms to temporally implicit schemes since it is difficult to know if a low-order implicit scheme is positivity-preserving. In the present paper, a new positivity-preserving algorithm is proposed in terms of the flux correction technique. And the factors of the flux correction are determined by a residual correction procedure. For a finite volume scheme that is capable of achieving a converged solution, we show that the correction factors are in the order of unity with additional high-order terms corresponding to the spatial and temporal rates of convergence. Therefore, the proposed positivity-preserving algorithm is accuracy-reserving and asymptotically consistent. The notable advantage of this method is that it does not rely on the existence of low-order positivity-preserving baseline schemes. Therefore, it can be applied to the implicit schemes solving Euler and especially Navier-Stokes equations. In the present paper, the proposed technique is applied to an implicit dual time-stepping finite volume scheme with temporal second-order and spatial high-order accuracy. The present positivity-preserving algorithm is implemented in an iterative manner to ensure that the dual time-stepping iteration will converge to the positivity-preserving solution. Another similar correction technique is also proposed to ensure that the solution remains positivity-preserving at each sub-iteration. Numerical results demonstrate that the proposed algorithm preserves positive density and internal energy in all test cases and significantly improves the robustness of the numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, we develop algorithms to overcome the curse of dimensionality in non-convex state-dependent Hamilton-Jacobi partial differential equations (HJ PDEs) arising from optimal control and differential game problems. The subproblems are independent and they can be implemented in an embarrassingly parallel fashion. This is ideal for perfect scaling in parallel computing. The algorithm is proposed to overcome the curse of dimensionality [1,2] when solving HJ PDE. The major contribution of the paper is to change either the solving of a PDE problem or an optimization problem over a space of curves to an optimization problem of a single vector, which goes beyond the work of [40]. We extend the method in [7,9,15] , and conjecture a (Lax-type) minimization principle to solve state-dependent HJ PDE when the Hamiltonian is convex, as well as a (Hopf-type) maximization principle to solve state-dependent HJ PDE when the Hamiltonian is non-convex , as a generalization of the well-known Hopf formula in [18,25,50]. We showed the validity of the formula under restricted assumption for the sake of completeness, and would like to bring our readers to [62] which validates our conjectures in a more general setting. We conjectured the weakest assumption of our formula to hold is a pseudoconvexity assumption similar to one stated in [50]. Our method is expected to have application in control theory, differential game problems and elsewhere. [ABSTRACT FROM AUTHOR]

Abstract Data Assimilation (DA) is an uncertainty quantification technique used to incorporate observed data into a prediction model in order to improve numerical forecasted results. Variational DA (VarDA) is based on the minimisation of a function which estimates the discrepancy between numerical results and observations. Operational forecasting requires real-time data assimilation. This mandates the choice of opportune methods to improve the efficiency of VarDA codes without loosing accuracy. Due to the scale of the forecasting area and the number of state variables used to describe the physical model, DA is a big data problem. In this paper, the Truncated Singular Value Decomposition (TSVD) is used to reduce the space dimension, alleviate the computational cost and reduce the errors. Nevertheless, a consequence is that important information is lost if the truncation parameter is not properly chosen. We provide an algorithm to compute the optimal truncation parameter and we prove that the optimal estimation reduces the ill-conditioning and removes the statistically less significant modes which could add noise to the estimate obtained from DA. In this paper, numerical issues faced in developing VarDA algorithm include the ill-conditioning of the background covariance matrix, the choice of a preconditioning and the choice of the regularisation parameter. We also show how the choice of the regularisation parameter impacts on the efficiency of the VarDA minimisation computed by the L-BFGS (Limited – Broyden Fletcher Goldfarb Shanno). Experimental results are provided for pollutant dispersion within an urban environment. Highlights • A Variational DA model is introduced for air flows prediction and pollution transport. • The optimal truncation parameter minimises condition number and Preconditioning Error. • The TSVD truncation parameter is independent from the knowledge of an exact solution. • The optimal truncation parameter does not require the whole spectrum of the matrix. [ABSTRACT FROM AUTHOR]

Abstract This paper addresses one of the main challenges of the conservative level set method, namely the ill-conditioned behavior of the normal vector away from the interface. An alternative formulation for reconstruction of the interface is proposed. Unlike commonly used methods, which rely on a unit normal vector, the Stabilized Conservative Level Set (SCLS) makes use of a modified normal vector with diminishing magnitude away from the interface. With the new formulation, in the vicinity of the interface the reinitialization procedure utilizes compressive flux and diffusive terms only in normal direction with respect to the interface, thus, preserving the conservative level set properties, while away from the interface the directional diffusion mechanism automatically switches to homogeneous diffusion. The proposed formulation is robust and general. It is especially well suited for use with the adaptive mesh refinement (AMR) approaches, since for computational accuracy high resolution is only required in the vicinity of the interface, while away from the interface low resolution simulations might be sufficient. All of the results reported in this paper are obtained using the Adaptive Wavelet Collocation Method, a general arbitrary order AMR-type method, which utilizes wavelet decomposition to adapt on steep gradients in the solution while retaining a predetermined order of accuracy. Numerical solution for a number of benchmark problems has been carried out to demonstrate the performance of the SCLS method. [ABSTRACT FROM AUTHOR]

This paper deals with the implementation of high order semi-Lagrangian particle methods to handle high dimensional Vlasov–Poisson systems. It is based on recent developments in the numerical analysis of particle methods and the paper focuses on specific algorithmic features to handle large dimensions. The methods are tested with uniform particle distributions in particular against a recent multi-resolution wavelet based method on a 4D plasma instability case and a 6D gravitational case. Conservation properties, accuracy and computational costs are monitored. The excellent accuracy/cost trade-off shown by the method opens new perspective for accurate simulations of high dimensional kinetic equations by particle methods. [ABSTRACT FROM AUTHOR]

This paper presents an automatic mesh generation algorithm for body-fitted structured meshes in Computational Fluids Dynamics (CFD) analysis using the Advancing Extraction Method (AEM). The method is applicable to two-dimensional domains with complex geometries, which have the hierarchical tree-like topography with extrusion-like structures (i.e., branches or tributaries) and intrusion-like structures (i.e., peninsula or dikes). With the AEM, the hierarchical levels of sub-domains can be identified, and the block boundary of each sub-domain in convex polygon shape in each level can be extracted in an advancing scheme. In this paper, several examples were used to illustrate the effectiveness and applicability of the proposed algorithm for automatic structured mesh generation, and the implementation of the method. [ABSTRACT FROM AUTHOR]

It is increasingly realized that taking stochastic effects into account is important in order to study biological cells. However, the corresponding mathematical formulation, the chemical master equation (CME), suffers from the curse of dimensionality and thus solving it directly is not feasible for most realistic problems. In this paper we propose a dynamical low-rank algorithm for the CME that reduces the dimensionality of the problem by dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error (everything else is computed exactly). This approach, compared to the commonly used stochastic simulation algorithm (SSA, a Monte Carlo method), has the advantage that it is completely noise-free. This is particularly important if one is interested in resolving the tails of the probability distribution. We show that in some cases (e.g. for the lambda phage) the proposed method can drastically reduce memory consumption and run time and provide better accuracy than SSA. • We propose a low-rank algorithm that does not rely on single orbital basis functions. • The proposed approach is better suited for biological/chemical reaction networks. • We demonstrate better accuracy and improved efficiency compared to SSA for some problems. • The proposed complexity reduction algorithm is noise free (in contrast to SSA). • We demonstrate that our algorithm accurately resolves the tails of the distribution (in contrast to SSA). [ABSTRACT FROM AUTHOR]

Steepest descent methods combining complex contour deformation with numerical quadrature provide an efficient and accurate approach for the evaluation of highly oscillatory integrals. However, unless the phase function governing the oscillation is particularly simple, their application requires a significant amount of a priori analysis and expert user input, to determine the appropriate contour deformation, and to deal with the non-uniformity in the accuracy of standard quadrature techniques associated with the coalescence of stationary points (saddle points) with each other, or with the endpoints of the original integration contour. In this paper we present a novel algorithm for the numerical evaluation of oscillatory integrals with general polynomial phase functions, which automates the contour deformation process and avoids the difficulties typically encountered with coalescing stationary points and endpoints. The inputs to the algorithm are simply the phase and amplitude functions, the endpoints and orientation of the original integration contour, and a small number of numerical parameters. By a series of numerical experiments we demonstrate that the algorithm is accurate and efficient over a large range of frequencies, even for examples with a large number of coalescing stationary points and with endpoints at infinity. As a particular application, we use our algorithm to evaluate cuspoid canonical integrals from scattering theory. A Matlab implementation of the algorithm is made available and is called PathFinder. • An algorithm for the evaluation of oscillatory integrals with polynomial phase. • The algorithm automates the numerical steepest descent method. • Our implementation is robust and efficient across all frequencies. [ABSTRACT FROM AUTHOR]

We propose an easy-to-implement iterative method for resolving the implicit (or semi-implicit) schemes arising in solving reaction-diffusion (RD) type equations. We formulate the nonlinear time implicit scheme as a min-max saddle point problem and then apply the primal-dual hybrid gradient (PDHG) method. Suitable precondition matrices are applied to the PDHG method to accelerate the convergence of algorithms under different circumstances. Furthermore, our method is applicable to various discrete numerical schemes with high flexibility. From various numerical examples tested in this paper, the proposed method converges properly and can efficiently produce numerical solutions with sufficient accuracy. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to constraining the eigenvalue range of symmetric tensors in numerical advection schemes based on the flux-corrected transport (FCT) algorithm and a continuous finite element discretization. In the context of element-based FEM-FCT schemes for scalar conservation laws, the numerical solution is evolved using local extremum diminishing (LED) antidiffusive corrections of a low order approximation which is assumed to satisfy the relevant inequality constraints. The application of a limiter to antidiffusive element contributions guarantees that the corrected solution remains bounded by the local maxima and minima of the low order predictor. The FCT algorithm to be presented in this paper guarantees the LED property for the maximal and minimal eigenvalues of the transported tensor at the low order evolution step. At the antidiffusive correction step, this property is preserved by limiting the antidiffusive element contributions to all components of the tensor in a synchronized manner. The definition of the element-based correction factors for FCT is based on perturbation bounds for auxiliary tensors which are constrained to be positive semidefinite to enforce the generalized LED condition. The derivation of sharp bounds involves calculating the roots of polynomials of degree up to 3. As inexpensive and numerically stable alternatives, limiting techniques based on appropriate estimates are considered. The ability of the new limiters to enforce local bounds for the eigenvalue range is confirmed by numerical results for 2D advection problems. [ABSTRACT FROM AUTHOR]

The MoF (Moment of Fluid) method is one of the most accurate approaches among various interface reconstruction algorithms. Alike other second order methods, the MoF method needs to solve an implicit optimization problem to obtain the optimal approximate interface, so an iteration process is inevitable under most circumstances. In order to solve the optimization efficiently, the properties of the objective function are worthy of studying. In 2D problems, the first order derivative has been deduced and applied in the previous researches. In this paper, the high order derivatives of the objective function are deduced on the convex polygon. We show that the n th ( n ≥ 2 ) order derivatives are discontinuous, and the number of the discontinuous points is two times the number of the polygon edge. A rotation algorithm is proposed to successively calculate these discontinuous points, thus the target interval where the optimal solution is located can be determined. Since the high order derivatives of the objective function are continuous in the target interval, the iteration schemes based on high order derivatives can be used to improve the convergence rate. Moreover, when iterating in the target interval, the value of objective function and its derivatives can be directly updated without explicitly solving the volume conservation equation. The direct update makes a further improvement of the efficiency especially when the number of edges of the polygon is increasing. The Halley's method, which is based on the first three order derivatives, is applied as the iteration scheme in this paper and the numerical results indicate that the CPU time is about half of the previous method on the quadrilateral cell and is about one sixth on the decagon cell. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modification of an acoustic-transport operator splitting Lagrange-projection method for simulating compressible flows with gravity. The original method involves two steps that respectively account for acoustic and transport effects. Our work proposes a simple modification of the transport step, and the resulting modified scheme turns out to be a flux-splitting method. This new numerical method is less computationally expensive in the low-Mach regime, more memory efficient, and easier to implement than the original one. We prove stability properties for this new scheme by showing that under classical CFL conditions, the method is positivity preserving for mass, energy and entropy satisfying. The flexible flux-splitting structure of the method enables straightforward extensions of the method to multi-dimensional problems (with respect to space) and high-order discretizations that are presented in this work. We also propose an interpretation of the flux-splitting solver as a relaxation approximation. Both the stability and the accuracy of the new method are tested against one-dimensional and two-dimensional numerical experiments that involve highly compressible flows and low-Mach regimes. • The all-regime acoustic/transport splitting solver proposed in [16] can be modified into a flux-splitting, one-step version. • The resulting method is less computationally expensive, more memory efficient, and easier to implement than the original one. • Stability properties are derived by interpreting the scheme as a convex combination of the advection/pressure steps. • This recasting allows an easy high-order extension as the new method is a standard flux-based solver. • Following the lines of [32] , we equip our scheme with the well-balanced property for gravity source terms. [ABSTRACT FROM AUTHOR]

For band-structure computation, the Bloch mode synthesis (BMS) or the generalized Bloch mode synthesis (GBMS) method is a series of model-reduction computational algorithms based on the Craig-Bampton method. A hybrid Bloch mode synthesis (HBMS) method with a hybrid Floquet-Bloch periodic boundary condition is proposed in this paper. A force-based Floquet-Bloch periodic boundary condition is derived and combined with the default displacement-based Floquet-Bloch periodic boundary condition to form the hybrid Floquet-Bloch periodic boundary condition. The hybrid BMS method has been developed by combining the hybrid Floquet-Bloch periodic boundary conditions with the free-interface component mode synthesis method. The HBMS method considers the influence of higher-order modes on periodic boundary structures so that the hybrid Floquet-Bloch periodic boundary conditions can be applied to the reduced-order modal space without introducing errors, which improves the computational efficiency. According to different truncation errors, the HBMS method has been developed into three algorithms, and they each have a different precision-efficiency ratio. Numerical experiments demonstrate that the developed method outperforms in accuracy and computational efficiency compared to the default BMS/GBMS method. • The force and displacement balance on the Bloch periodic boundary are considered. • Higher-order modes are considered by introducing force Bloch boundary conditions. • The Bloch boundary conditions are directly imposed on the reduced order model. • A new hybrid Bloch mode synthesis method for phononic was developed. • The proposed method is developed into three algorithms for practicality. [ABSTRACT FROM AUTHOR]

This paper concerns the cell-boundary error present in multiscale algorithms for elliptic homogenization problems. Typical multiscale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. To solve the micro model, boundary conditions are required on the boundary of the microscopic domain. Imposing a naive boundary condition leads to O ( ε / η ) error in the computation, where ε is the size of the microscopic variations in the media and η is the size of the micro-domain. The removal of this error in modern multiscale algorithms still remains an important open problem. In this paper, we present a time-dependent approach which is general in terms of dimension. We provide a theorem which shows that we have arbitrarily high order convergence rates in terms of ε / η in the periodic setting. Additionally, we present numerical evidence showing that the method improves the O ( ε / η ) error to O ( ε ) in general non-periodic media. [ABSTRACT FROM AUTHOR]

Game theory and its particular Nash Equilibrium (NE) are gaining importance in solving Multi Criterion Optimization (MCO) in engineering problems over the past decade. The solution of a MCO problem can be viewed as a NE under the concept of competitive games. This paper surveyed/proposed four efficient algorithms for calculating a NE of a MCO problem. Existence and equivalence of the solution are analyzed and proved in the paper based on fixed point theorem. Specific virtual symmetric Nash game is also presented to set up an optimization strategy for single objective optimization problems. Two numerical examples are presented to verify proposed algorithms. One is mathematical functions' optimization to illustrate detailed numerical procedures of algorithms, the other is aerodynamic drag reduction of civil transport wing fuselage configuration by using virtual game. The successful application validates efficiency of algorithms in solving complex aerodynamic optimization problem. [ABSTRACT FROM AUTHOR]

This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier–Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas–Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are “quasi-unconditionally stable” in the following sense: each algorithm is stable for all couples ( h , Δ t ) of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form ( 0 , M h ) × ( 0 , M t ) . In other words, for each fixed value of Δ t below a certain threshold, the Navier–Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second-order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier–Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier–Stokes solvers for which second-order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear coarse-mesh finite difference method with two-node local problem (CMFD2N) is proven to be unconditionally stable for neutron diffusion eigenvalue problems. The explicit current correction factor (CCF) is derived based on the two-node analytic nodal method (ANM2N), and a Fourier stability analysis is applied to the linearized algorithm. It is shown that the analytic convergence rate obtained by the Fourier analysis compares very well with the numerically measured convergence rate. It is also shown that the theoretical convergence rate is only governed by the converged second harmonic buckling and the mesh size. It is also noted that the convergence rate of the CCF of the CMFD2N algorithm is dependent on the mesh size, but not on the total problem size. This is contrary to expectation for eigenvalue problem. The novel points of this paper are the analytical derivation of the convergence rate of the CMFD2N algorithm for eigenvalue problem, and the convergence analysis based on the analytic derivations. [ABSTRACT FROM AUTHOR]

In this paper we introduce an efficient algorithm for identifying conductive objects using induction data derived from eddy currents. Our method consists of first extracting geometric features from the induction data and then matching them to precomputed data for known objects from a given dictionary. The matching step relies on fundamental properties of conductive polarization tensors and new invariance properties introduced in this paper. A new shape identification scheme is developed and tested in numerical simulations in the presence of measurement noise. Resolution and stability properties of the proposed identification algorithm are investigated. [ABSTRACT FROM AUTHOR]

The objective of this paper is the exposition of a recently-developed, novel Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a cylindrical conducting object, carrying a potential and moving at low speeds through an otherwise neutral medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional Monte Carlo based approaches for the solution of nonlinear equations are iterative in nature, involving branching stochastic processes which are used to calculate linear functionals of the solution of nonlinear integral equations. Over the last several years, one of the authors of this paper, K. Chatterjee has been developing a philosophically-different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration and the simulation of branching processes. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains where the random walker makes its walks. However, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. The application area of interest is in the modeling of the communication breakdown problem during a space vehicle's re-entry into the atmosphere. However, additional application areas are being explored in the modeling of electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications. [ABSTRACT FROM AUTHOR]

This paper presents a stable numerical algorithm for the Brinkman equations by using weak Galerkin (WG) finite element methods. The Brinkman equations can be viewed mathematically as a combination of the Stokes and Darcy equations which model fluid flow in a multi-physics environment, such as flow in complex porous media with a permeability coefficient highly varying in the simulation domain. In such applications, the flow is dominated by Darcy in some regions and by Stokes in others. It is well known that the usual Stokes stable elements do not work well for Darcy flow and vice versa. The challenge of this study is on the design of numerical schemes which are stable for both the Stokes and the Darcy equations. This paper shows that the WG finite element method is capable of meeting this challenge by providing a numerical scheme that is stable and accurate for both Darcy and the Stokes dominated flows. Error estimates of optimal order are established for the corresponding WG finite element solutions. The paper also presents some numerical experiments that demonstrate the robustness, reliability, flexibility and accuracy of the WG method for the Brinkman equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a second-order numerical method for simulations of reacting flow around heat-conducting immersed solid objects. The method is coupled with a block-structured adaptive mesh refinement (SAMR) framework and a low-Mach number operator-split projection algorithm. A “buffer zone” methodology is introduced to impose the solid–fluid boundary conditions such that the solver uses symmetric derivatives and interpolation stencils throughout the interior of the numerical domain; irrespective of whether it describes fluid or solid cells. Solid cells are tracked using a binary marker function. The no-slip velocity boundary condition at the immersed wall is imposed using the staggered mesh. Near the immersed solid boundary, single-sided buffer zones (inside the solid) are created to resolve the species discontinuities, and dual buffer zones (inside and outside the solid) are created to capture the temperature gradient discontinuities. The development discussed in this paper is limited to a two-dimensional Cartesian grid-conforming solid. We validate the code using benchmark simulations documented in the literature. We also demonstrate the overall second-order convergence of our numerical method. To demonstrate its capability, a reacting flow simulation of a methane/air premixed flame stabilized on a channel-confined bluff-body using a detailed chemical kinetics model is discussed. [Copyright &y& Elsevier]

This paper presents two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation, while most existing structure-preserving algorithms are only valid for zero or periodic boundary conditions. The first strategy is based on the conventional second-order central difference quotient but with a cell-centered grid, while the other is established on the regular grid but incorporated with summation by parts (SBP) operators. Both the methodologies can provide conservative semi-discretizations with different forms of Hamiltonian structures and the discrete energy. However, utilizing the existing SBP formulas, schemes obtained by the second strategy can directly achieve higher-order accuracy while it is not obvious for schemes based on the cell-centered grid to make accuracy improved easily. Further combining the implicit midpoint method and the scalar auxiliary variable (SAV) approach, we construct symplectic integrators and linearly implicit energy-preserving schemes for the two-dimensional sine-Gordon equation, respectively. Extensive numerical experiments demonstrate their effectiveness with the homogeneous Neumann boundary conditions. • Two kinds of structure-preserving algorithms with Neumann boundary condition are proposed. • High-order schemes can be easily obtained in the framework of summation-by-parts operators. • The semi-discrete energy and the symplectic structure are still valid. • Linearly implicit energy-preserving schemes are proposed based on the SAV approach. • Such methodology can be generalized to other kinds of conservative systems with Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the problem of generating independent samples from a conditional distribution when independent samples from the prior distribution are available. Although there are exact methods for sampling from the posterior (e.g. Markov chain Monte Carlo or acceptance/rejection), these methods tend to be computationally demanding when evaluation of the likelihood function is expensive, as it is for most geoscience applications. As an alternative, in this paper we discuss deterministic mappings of variables distributed according to the prior to variables distributed according to the posterior. Although any deterministic mappings might be equally useful, we will focus our discussion on a class of algorithms that obtain implicit mappings by minimization of a cost function that includes measures of data mismatch and model variable mismatch. Algorithms of this type include quasi-linear estimation, randomized maximum likelihood, perturbed observation ensemble Kalman filter, and ensemble of perturbed analyses (4D-Var). When the prior pdf is Gaussian and the observation operators are linear, we show that these minimization-based simulation methods solve an optimal transport problem with a nonstandard cost function. When the observation operators are nonlinear, however, the mapping of variables from the prior to the posterior obtained from those methods is only approximate. Errors arise from neglect of the Jacobian determinant of the transformation and from the possibility of discontinuous mappings. [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]

• Designed coupled implicit-explicit temporal and spatial splitting method. • Formulated conditions for stability and designed appropriate spaces. • Multiscale spaces with CEM-GMsFEM are used in constructing the splitting method. • Numerical results confirm our theoretical findings. Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extent are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is d t = H 2 / κ m a x , where κ m a x is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The contrast-independent stability of these algorithms requires a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented. [ABSTRACT FROM AUTHOR]

Abstract: We propose a dynamically bi-orthogonal method (DyBO) to solve time dependent stochastic partial differential equations (SPDEs). The objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi-orthogonal basis. It is well-known that the Karhunen–Loeve expansion (KLE) minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. The main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the covariance matrix or to compute its eigendecomposition. In the first part of our paper, we introduce the derivation of the dynamically bi-orthogonal formulation for SPDEs, discuss several theoretical issues, such as the dynamic bi-orthogonality preservation and some preliminary error analysis of the DyBO method. We also give some numerical implementation details of the DyBO methods, including the representation of stochastic basis and techniques to deal with eigenvalue crossing. In the second part of our paper [11], we will present an adaptive strategy to dynamically remove or add modes, perform a detailed complexity analysis, and discuss various generalizations of this approach. An extensive range of numerical experiments will be provided in both parts to demonstrate the effectiveness of the DyBO method. [Copyright &y& Elsevier]

Abstract: This is part II of our paper in which we propose and develop a dynamically bi-orthogonal method (DyBO) to study a class of time-dependent stochastic partial differential equations (SPDEs) whose solutions enjoy a low-dimensional structure. In part I of our paper [9], we derived the DyBO formulation and proposed numerical algorithms based on this formulation. Some important theoretical results regarding consistency and bi-orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the DyBO method. In this paper, we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically. Our complexity analysis shows that the ratio of computational complexities between the DyBO method and a generalized polynomial chaos method (gPC) is roughly of order for a quadratic nonlinear SPDE, where m is the number of mode pairs used in the DyBO method and is the number of elements in the polynomial basis in gPC. The effective dimensions of the stochastic solutions have been found to be small in many applications, so we can expect m is much smaller than and computational savings of our DyBO method against gPC are dramatic. The adaptive strategy plays an essential role for the DyBO method to be effective in solving some challenging problems. Another important contribution of this paper is the generalization of the DyBO formulation for a system of time-dependent SPDEs. Several numerical examples are provided to demonstrate the effectiveness of our method, including the Navier–Stokes equations and the Boussinesq approximation with Brownian forcing. [Copyright &y& Elsevier]

Abstract: When one solves PDEs modelling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. For instance, the underlying physical assumptions for the Euler equations are the positivity of both density and pressure variables. We consider in this paper an unstructured vertex-based tesselation in . Given a MUSCL finite volume scheme and given a reconstruction method (including a limiting process), the point is to determine whether the overall scheme ensures the positivity. The present work is issued from seminal papers from Perthame and Shu (On positivity preserving finite volume schemes for Euler equations, Numer. Math. 73 (1996) 119–130) and Berthon (Robustness of MUSCL schemes for 2D unstructured meshes, J. Comput. Phys. 218 (2) (2006) 495–509). They proved in different frameworks that under assumptions on the corresponding one-dimensional numerical flux, a suitable CFL condition guarantees that density and pressure remain positive. We first analyse Berthon’s method by presenting the ins and outs. We then propose a more general approach adding non geometric degrees of freedom. This approach includes an optimization procedure in order to make the CFL condition explicit and as less restrictive as possible. The reconstruction method is handled independently by means of τ-limiters and of an additional damping parameter. An algorithm is provided in order to specify the adjustments to make in a preexisting code based on a certain numerical flux. Numerical simulations are carried out to prove the accuracy of the method and its ability to deal with low densities and pressures. [Copyright &y& Elsevier]

Abstract: This paper describes a unified, element based Galerkin (EBG) framework for a three-dimensional, nonhydrostatic model for the atmosphere. In general, EBG methods possess high-order accuracy, geometric flexibility, excellent dispersion properties and good scalability. Our nonhydrostatic model, based on the compressible Euler equations, is appropriate for both limited-area and global atmospheric simulations. Both a continuous Galerkin (CG), or spectral element, and discontinuous Galerkin (DG) model are considered using hexahedral elements. The formulation is suitable for both global and limited-area atmospheric modeling, although we restrict our attention to 3D limited-area phenomena in this study; global atmospheric simulations will be presented in a follow-up paper. Domain decomposition and communication algorithms used by both our CG and DG models are presented. The communication volume and exchange algorithms for CG and DG are compared and contrasted. Numerical verification of the model was performed using two test cases: flow past a 3D mountain and buoyant convection of a bubble in a neutral atmosphere; these tests indicate that both CG and DG can simulate the necessary physics of dry atmospheric dynamics. Scalability of both methods is shown up to 8192 CPU cores, with near ideal scaling for DG up to 32,768 cores. [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: A systematic approach is presented for constructing higher-order immersed boundary and ghost fluid methods for CFD in general, and fluid–structure interaction problems in particular. Such methods are gaining popularity because they simplify a number of computational issues. These range from gridding the fluid domain, to designing and implementing Eulerian-based algorithms for challenging fluid–structure applications characterized by large structural motions and deformations or topological changes. However, because they typically operate on non body-fitted grids, immersed boundary and ghost fluid methods also complicate other issues such as the treatment of wall boundary conditions in general, and fluid–structure transmission conditions in particular. These methods also tend to be at best first-order space-accurate at the immersed interfaces. In some cases, they are also provably inconsistent at these locations. A methodology is presented in this paper for addressing this issue. It is developed for inviscid flows and prescribed structural motions. For the sake of clarity, but without any loss of generality, this methodology is described in one and two dimensions. However, its extensions to flow-induced structural motions and three dimensions are straightforward. The proposed methodology leads to a departure from the current practice of populating ghost fluid values independently from the chosen spatial discretization scheme. Instead, it accounts for the pattern and properties of a preferred higher-order discretization scheme, and attributes ghost values as to preserve the formal order of spatial accuracy of this scheme. It is illustrated in this paper by its application to various finite difference and finite volume methods. Its impact is also demonstrated by one- and two-dimensional numerical experiments that confirm its theoretically proven ability to preserve higher-order spatial accuracy, including in the vicinity of the immersed interfaces. [Copyright &y& Elsevier]

Abstract: This paper presents an overview and introduction to smoothed particle hydrodynamics and magnetohydrodynamics in theory and in practice. Firstly, we give a basic grounding in the fundamentals of SPH, showing how the equations of motion and energy can be self-consistently derived from the density estimate. We then show how to interpret these equations using the basic SPH interpolation formulae and highlight the subtle difference in approach between SPH and other particle methods. In doing so, we also critique several ‘urban myths’ regarding SPH, in particular the idea that one can simply increase the ‘neighbour number’ more slowly than the total number of particles in order to obtain convergence. We also discuss the origin of numerical instabilities such as the pairing and tensile instabilities. Finally, we give practical advice on how to resolve three of the main issues with SPMHD: removing the tensile instability, formulating dissipative terms for MHD shocks and enforcing the divergence constraint on the particles, and we give the current status of developments in this area. Accompanying the paper is the first public release of the ndspmhd SPH code, a 1, 2 and 3 dimensional code designed as a testbed for SPH/SPMHD algorithms that can be used to test many of the ideas and used to run all of the numerical examples contained in the paper. [Copyright &y& Elsevier]

Abstract: In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gas–droplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N, N >2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N-dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the Eulerian Multi Size Moment (EMSM) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor–Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed. [Copyright &y& Elsevier]

This paper presents an accelerated quadrature scheme for the evaluation of layer potentials in three dimensions. Our scheme combines a generic, high order quadrature method for singular kernels called Quadrature by Expansion (QBX) with a modified version of the Fast Multipole Method (FMM). Our scheme extends a recently developed formulation of the FMM for QBX in two dimensions, which, in that setting, achieves mathematically rigorous error and running time bounds. In addition to generalization to three dimensions, we highlight some algorithmic and mathematical opportunities for improved performance and stability. Lastly, we give numerical evidence supporting the accuracy, performance, and scalability of the algorithm through a series of experiments involving the Laplace and Helmholtz equations. • A new fast algorithm for Quadrature by Expansion (QBX) in three dimensions. • Comprehensive strategies to control truncation, quadrature, and acceleration error. • Empirical results that confirm expected error and scaling behavior. [ABSTRACT FROM AUTHOR]

Abstract The level set method for modeling multi-fluid interfaces gives a simple approach for the simulation of multiphase flow problems. However, many of the components of this method, such as the redistancing of the level set function and discretization of the singular surface tension force, become more difficult on unstructured meshes in a control volume framework; furthermore, these operations are shown to lead to errors in conservation of mass of the individual phases, as well as unphysical currents near the interface. This paper presents a novel formulation for the level set method combined with a balanced-force algorithm using a control volume finite element method (CVFEM) for unstructured grids. A diffuse interface description is used, allowing a straightforward inclusion of surface tension forces; we show that by consistently evaluating the pressure gradient and surface tension, we are able to discretely balance these two forces and suppress any parasitic currents given a prescribed curvature. By converting the redistancing equation to a form that is suitable for CVFEM and enforcing a constraint directly linked with the phase fraction in each cell, we are able conserve mass and preserve the shape of the interface to a high degree of accuracy in multiphase flow problems with high density and viscosity ratios. The accuracy of the of our proposed method is assessed by comparing with benchmark results from the literature. Highlights • A novel level set formulation utilizing CVFEM is proposed. • A balanced-force algorithm is formulated for purely unstructured meshes. • The redistancing equation is reformulated to allow higher order upwinding methods. • A new constraint is introduced to preserve the shape of the interface. • Benchmark examples show convergence for high density and viscosity ratio problems. [ABSTRACT FROM AUTHOR]

Abstract Many codes have been developed to study highly relativistic, magnetized flows around and inside compact objects. Depending on the adopted formalism, some of these codes evolve the vector potential A , and others evolve the magnetic field B = ∇ × A directly. Given that these codes possess unique strengths, sometimes it is desirable to start a simulation using a code that evolves B and complete it using a code that evolves A. Thus transferring the data from one code to another would require an inverse curl algorithm. This paper describes two new inverse curl techniques in the context of Cartesian numerical grids: a cell-by-cell method, which scales approximately linearly with the numerical grid, and a global linear algebra approach, which has worse scaling properties but is generally more robust (e.g., in the context of a magnetic field possessing some nonzero divergence). We demonstrate these algorithms successfully generate smooth vector potential configurations in challenging special and general relativistic contexts. Highlights • Our solution consists of a numerical implementation of the "inverse curl" operator. • We use two different methods that have different strengths and weaknesses. • The cell-by-cell technique is very fast and scales with the number of gridpoints. • The global linear algebra method is slower but has better symmetry properties. [ABSTRACT FROM AUTHOR]

Abstract In cylindrical/spherical geometries, MHD solvers using explicit finite-volume methods face restrictive time steps imposed by the CFL condition due to the clustering of cells in the azimuthal direction near the pole/axis. We use a conservative averaging-reconstruction method (Ring Average) on structured cylindrical/spherical mesh to remove this severe time step restriction for multi-dimensional curvilinear finite-volume MHD solvers. The Ring Average technique is implemented as a post-processing step and thus requires no changes to the existing data structure, grid definition or numerical methods in the original MHD solver. This paper describes the Ring Average algorithm and presents simulation results using field loop advection and MHD blast waves in cylindrical/spherical geometries for validation. The algorithm is shown to be inexpensive and easily implemented in existing curvilinear finite-volume MHD solvers. Highlights • Ring Average is a "post-processing" module without modifying the original solver. • Ring Average relaxes time-step restrictions in MHD solvers with axis singularities. • Ring Average conserves mass, momentum and energy for proper shock propagations. • Ring Average works for codes with either divergence cleaning or constrained transport. [ABSTRACT FROM AUTHOR]

Abstract: Modeling of laser-plasma wakefield accelerators in an optimal frame of reference has been shown to produce orders of magnitude speed-up of calculations from first principles. Obtaining these speedups required mitigation of a high-frequency instability that otherwise limits effectiveness. In this paper, methods are presented which mitigated the observed instability, including an electromagnetic solver with tunable coefficients, its extension to accommodate Perfectly Matched Layers and Friedman’s damping algorithms, as well as an efficient large bandwidth digital filter. It is observed that choosing the frame of the wake as the frame of reference allows for higher levels of filtering or damping than is possible in other frames for the same accuracy. Detailed testing also revealed the existence of a singular time step at which the instability level is minimized, independently of numerical dispersion. A combination of the techniques presented in this paper prove to be very efficient at controlling the instability, allowing for efficient direct modeling of 10GeV class laser plasma accelerator stages. The methods developed in this paper may have broader application, to other Lorentz-boosted simulations and Particle-In-Cell simulations in general. [Copyright &y& Elsevier]

Abstract This paper develops a new implementation coupling optimisation-based anisotropic mesh adaptivity algorithms to a moving mesh numerical scour model, considering both turbulent suspended and bedload sediment transport. The significant flexibility over mesh structure and resolution, in space and time, that the coupling of these approaches provides makes this framework highly suitable for resolving individual marine structure scales with larger scale ocean dynamics. The use of mesh optimisation addresses the issue of poor mesh quality and/or inappropriate resolution that have compromised existing modelling approaches that apply mesh movement strategies alone, especially in the case of extreme scour. Discontinuous Galerkin finite element-based discretisation methods and a Reynolds Averaged Navier–Stokes-based turbulent modelling approach are used for the hydrodynamic fluid flow. In this work the model is verified in two dimensions for current-dominated scour near a horizontal pipeline. Combined adaptive mesh movement and anisotropic mesh optimisation is found to maintain both the quality and validity of the mesh in response to morphological bed evolution changes, even in the case where it is severely constrained by nearby structures. [ABSTRACT FROM AUTHOR]

Abstract In scalar turbulence it is sometimes the case that the scalar diffusivity is smaller than the viscous diffusivity. The thermally-driven turbulent convection in water is a typical example. In such a case the smallest scale in the problem is the Batchelor scale l b , rather than the Kolmogorov scale l k , as l b = l k / S c 1 / 2 , where Sc is the Schmidt number (or Prandtl number in the case of temperature). In the numerical studies of such scalar turbulence, the conventional approach is to use a single grid for both the velocity and scalar fields. Such single-resolution scheme often over-resolves the velocity field because the resolution requirement for scalar is higher than that for the velocity field, since l b < l k for S c > 1. In this paper we put forward an algorithm that implements the so-called multiple-resolution method with a finite-volume code. In this scheme, the velocity and pressure fields are solved in a coarse grid, while the scalar field is solved in a dense grid. The central idea is to implement the interpolation scheme on the framework of finite-volume to reconstruct the divergence-free velocity from the coarse to the dense grid. We demonstrate our method using a canonical model system of fluid turbulence, the Rayleigh–Bénard convection. We show that, with the tailored mesh design, considerable speed-up for simulating scalar turbulence can be achieved, especially for large Schmidt (Prandtl) numbers. In the same time, sufficient accuracy of the scalar and velocity fields can be achieved by this multiple-resolution scheme. Although our algorithm is demonstrated with a case of an active scalar, it can be readily applied to passive scalar turbulent flows. [ABSTRACT FROM AUTHOR]

Abstract Large computation time is widely considered to be the most important issue in scientific research especially in solving structural evolution problems. Recent developments in this domain have shown that the use of non-incremental schemes through Model Order Reduction led to important results in saving time. Yet, the question arises here how to obtain more time-saving. This paper examines an approach based on a collection of significant modes given by Proper Generalized Decomposition (PGD) solution for different time scales in order to save more computation time. The dictionary of the significant modes allows to construct an accurate solution for different characteristic times and different boundary problems compared to the full solution with a relative error rate less than 5% and with a large time saving of order 50 compared to Finite Element Method (FEM). The ability of the approach with respect to cycle time is discussed. Highlights • A new approach is proposed to decrease computational time for cyclic problems involving different time scales. • The approach is based on the use of pre-computed space–time modes which are further collected in a dictionary. • For a physical time belonging to the range of the dictionary, the solution can be accurately predicted. • For a given cycle time, the dictionary leads to an accurate if it contains the modes associated with this specific time. [ABSTRACT FROM AUTHOR]

Abstract We present a systematic study of the nested sampling algorithm based on the example of the Potts model. This model, which exhibits a first order phase transition for q > 4 , exemplifies a generic numerical challenge in statistical physics: The evaluation of the partition function and thermodynamic observables, which involve high dimensional sums of sharply structured multi-modal density functions. It poses a major challenge to most standard numerical techniques, such as Markov Chain Monte Carlo. In this paper we will demonstrate that nested sampling is particularly suited for such problems and it has a couple of advantages. For calculating the partition function of the Potts model with N sites: a) one run stops after O (N) moves, so it takes O (N 2) operations for the run, b) only a single run is required to compute the partition function along with the assignment of confidence intervals, c) the confidence intervals of the logarithmic partition function decrease with 1 / N and d) a single run allows to compute quantities for all temperatures while the autocorrelation time is very small, irrespective of temperature. Thermodynamic expectation values of observables, which are completely determined by the bond configuration in the representation of Fortuin and Kasteleyn, like the Helmholtz free energy, the internal energy as well as the entropy and heat capacity, can be calculated in the same single run needed for the partition function along with their confidence intervals. In contrast, thermodynamic expectation values of magnetic properties like the magnetization and the magnetic susceptibility require sampling the additional spin degree of freedom. Results and performance are studied in detail and compared with those obtained with multi-canonical sampling. Eventually the implications of the findings on a parallel implementation of nested sampling are outlined. [ABSTRACT FROM AUTHOR]

Highlights • Space-time finite element discretizations for constrained FOSLS are studied numerically for time-dependent PDEs. • The considered formulation provides the expected orders of approximation and convergence. • A multilevel algorithm can be used to convert the saddle-point problem into the divergence-free setting to use multigrid. • Two different preconditioning strategies are studied and compared numerically. • Multigrid preconditioners in the div-free space outperform the block-diagonal preconditioners for the saddle point systems. Abstract This paper studies finite element discretizations for three types of time-dependent PDEs, namely heat equation, scalar conservation law and wave equation, which we reformulate as first order systems in a least-squares setting, subject to a space-time conservation constraint (coming from the original PDE). Available piecewise polynomial finite element spaces in (n + 1) -dimensions for functional spaces from the (n + 1) -dimensional de Rham sequence for n = 2 , 3 are used for the implementation of the method. Computational results illustrating the error behavior, iteration counts and performance of block-diagonal and monolithic geometric multigrid preconditioners are presented for the discrete CFOSLS system. The results are obtained from a parallel implementation of the methods for which we report reasonable scalability. [ABSTRACT FROM AUTHOR]

Highlights • Volume of fluid method with high-order representation for moving interface. • Rigorous conservation of mass/volume of the transported VOF field. • Sub-grid resolution for fine interfacial structures that are smaller than mesh size. • Easy-to-code and straightforward algorithm for 3D unstructured-grid. • Verified superior solution quality in comparison with classical VOF methods. Abstract In this paper, we propose a simple and accurate numerical method for capturing moving interfaces on fixed Eulerian grids by coupling the Tangent of Hyperbola Interface Capturing (THINC) method and Level Set (LS) method. The innovative and practically-significant aspects of the proposed method, so-called THINC/LS method, lie in (1) representing the interface with polynomial of high-order (arbitrary order in principle) rather than the plane representation commonly used in the Piecewise Linear Interface Calculation (PLIC) Volume of fluid (VOF) methods, (2) conserving rigorously the mass of the transported VOF field, (3) being able to resolving fine interface structures under mesh resolution, and (4) providing a straightforward and easy-to-code algorithm for 3D implementation. We verified the proposed scheme with the widely used benchmark tests. Numerical results show that this new method can obtain high-order accuracy for interface reconstruction and can produce more accurate results in capturing moving interfaces compared to the classical VOF methods. [ABSTRACT FROM AUTHOR]

Abstract This paper describes a novel partitioned algorithm for fluid–structure interaction (FSI) problems that couples the motion of rigid bodies and incompressible flow. This is the first partitioned algorithm that remains stable and second-order accurate, without sub-time-step iterations, for very light, and even zero-mass, bodies in three dimensions. This new added-mass partitioned (AMP) algorithm extends the previous developments in [1,2] by generalizing the added-damping tensors to account for arbitrary three-dimensional rotations, and by employing a general quadrature for the surface integral over a rigid body to derive the discrete AMP interface condition for the fluid pressure. Stability analyses for two three-dimensional model problems show that the algorithm remains stable for bodies of any mass when applied to the relevant model problems. The resulting AMP algorithm is implemented in parallel using a moving composite grid framework to treat one or more rigid bodies in complex three-dimensional configurations. The new three-dimensional algorithm is verified and validated though several benchmark problems, including the motion of a sphere in a viscous incompressible fluid and the interaction of a bi-leaflet mechanical heart valve and a pulsating fluid. Numerical simulations confirm the predictions of the stability analysis even for complex problems, and show that the AMP algorithm remains stable, without sub-iterations, for light and even zero-mass three-dimensional rigid bodies of general shape. These benchmark problems are further used to examine the parallel performance of the algorithm and to investigate the conditioning of the linear system for the pressure including the newly derived AMP interface conditions. Highlights • A partitioned FSI scheme for incompressible flow and rigid bodies is extended to 3D. • Stability for simple geometries is proved using 3D model problems. • Numerical results confirm stability and accuracy for light bodies of general shapes. • A benchmark for a bi-leaflet mechanical heart valve is designed and conducted. • Refinement studies for the heart valve are provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new heterogeneous fast multipole method (H-FMM) for 2-D Helmholtz equation in layered media. To illustrate the main algorithm ideas, we focus on the case of two layers in this work. The key compression step in the H-FMM is based on a fact that the multipole expansion for the sources of the free-space Green's function can be used also to compress the far field of the sources of the layered-media or domain Green's function, and a similar result exists for the translation operators for the multipole and local expansions. The mathematical error analysis is shown rigorously by an image representation of the Sommerfeld spectral form of the domain Green's function. As a result, in the H-FMM algorithm, the multipole-to-multipole, multipole-to-local, and local-to-local translation operators are the same as those in the free-space case, allowing easy adaptation of existing free-space FMM. All the spatially variant information of the domain Green's function is collected into the multipole-to-local translations and therefore the FMM becomes heterogeneous. The compressed representation further reduces the cost of evaluating the domain Green's function when computing the local direct interactions. Preliminary numerical experiments are presented to demonstrate the efficiency and accuracy of the algorithm with much improved performance over some existing methods for inhomogeneous media. [ABSTRACT FROM AUTHOR]

Uncertainty quantification is an emerging research area aiming at quantifying the variation in engineering system outputs due to uncertain inputs. One approach to study problems in uncertainty quantification is using polynomial chaos expansions. Though, a well-known limitation of polynomial chaos approaches is that their computational cost becomes prohibitive when the dimension of the stochastic space is large. In this paper, we propose a procedure to solve high dimensional stochastic problems with a limited computational budget. The methodology is based on an existing non-intrusive model reduction scheme for polynomial chaos representation, introduced by Raisee et al. [1] , that is further extended by introducing sparse polynomial chaos expansions. Specifically, an optimal stochastic basis is calculated from a coarse scale analysis, using proper orthogonal decomposition and sparse polynomial chaos and is then utilized in the fine scale analysis. This way, the computational expense on both the coarse and fine discretization levels is drastically reduced. Two application examples are considered to validate the proposed method and demonstrate its potential in solving high dimensional uncertainty quantification problems. One analytical stochastic problems is first studied, where up to 20 uncertainties were introduced in order to challenge the proposed method. A more realistic CFD type application is then discussed. It consists of a two dimensional NACA 0012 symmetric profile operating at subsonic flight conditions. It is shown that the proposed reduced order method based on sparse polynomial chaos expansions is able to predict statistical quantities with little loss of information, at a cheaper cost than other state-of-the-art techniques. [ABSTRACT FROM AUTHOR]

Centroidal Voronoi tessellation (CVT)-based mesh generation is a very effective technique for creating high-quality Voronoi meshes and their dual Delaunay triangulations that often play a crucial role in applications, including ocean and atmospheric simulations using finite volume schemes. In the next generation climate models, the spacing scales change dramatically across the whole sphere and require ultra-high resolution and smooth transitions from coarse to fine grid regions. Thus fast and robust spherical CVT (SCVT) meshing algorithms become highly desirable. In this paper, we first propose a Lloyd-preconditioned limited-memory BFGS method for constructing SCVTs that is also applicable to the construction of CVTs of general domains. This method is then parallelized based on overlapping domain decomposition, enabling excellent scalability on distributed systems. Results of several computational experiments show that the new method could incur computational time costs one order of magnitude smaller compared with some existing methods for generating large-scale highly variable-resolution meshes, while also providing significant improvements in mesh quality. [ABSTRACT FROM AUTHOR]

Abstract: High-order methods that can resolve interactions of flow-disturbances with shock waves are critical for reliable numerical simulation of shock wave and turbulence interaction. Such problems are not well understood due to the limitations of numerical methods. Most of the popular shock-capturing methods are only first-order accurate at the shock and may incur spurious numerical oscillations near the shock. Shock-fitting algorithms have been proposed as an alternative which can achieve uniform high-order accuracy and can avoid possible spurious oscillations incurred in shock-capturing methods by treating shocks as sharp interfaces. We explore two ways for shock-fitting: conventional moving grid set-up and a new fixed grid set-up with front tracking. In the conventional shock-fitting method, a moving grid is fitted to the shock whereas in the newly developed fixed grid set-up the shock front is tracked using Lagrangian points and is free to move across the underlying fixed grid. Different implementations of shock-fitting methods have been published in the literature. However, uniform high-order accuracy of various shock-fitting methods has not been systematically established. In this paper, we carry out a rigorous grid-convergence analysis on different variations of shock-fitting methods with both moving and fixed grids. These shock-fitting methods consist of different combinations of numerical methods for computing flow away from the shock and those for computing the shock movement. Specifically, we consider fifth-order upwind finite-difference scheme and shock-capturing WENO schemes with conventional shock-fitting and show that a fifth-order convergence is indeed achieved for a canonical one-dimensional shock-entropy wave interaction problem. We also show that the method of finding shock velocity from one characteristic relation and Rankine–Hugoniot jump condition performs better than the other methods of computing shock velocities. A high-order front-tracking implementation of shock-fitting is also presented in this paper and nominal rate of convergence is shown. The front-tracking results are validated by comparing to results from the conventional shock-fitting method and a linear-interaction analysis for a two-dimensional shock disturbance interaction problem. [Copyright &y& Elsevier]

Abstract: This paper describes the algorithms and functionality of a new module developed to support overset grid assembly associated with performing time-dependent and adaptive moving body calculations of external aerodynamic flows using a multi-solver paradigm (i.e. different CFD solvers in different parts of the computational domain). We use the term “domain connectivity” in this paper to denote all the procedures that are involved in an overset grid assembly, and the module developed is referred henceforth as the domain-connectivity module. The domain-connectivity module coordinates the data transfer between different solvers applied in different parts of the computational domain – body fitted structured or unstructured to capture viscous near-wall effects, and Cartesian adaptive mesh refinement to capture effects away from the wall. The execution of the CFD solvers and the domain-connectivity module are orchestrated by a Python-based computational infrastructure. The domain-connectivity module is fully parallel and performs all its operations (identification of grid overlaps and determination of data interpolation strategy) on the partitioned grid data. In addition, the domain connectivity procedures are completely automated such that no user intervention or manual input is necessary. The capabilities and performance of the package are presented for several test problems, including flow over a NACA 0015 wing and an AGARD A2 slotted airfoil, hover simulation of a scaled V-22 rotor, and dynamic simulation of a UH-60A rotor in forward flight. A modification to the algorithm for improved domain connectivity solutions in problems with tight tolerances as well as heterogeneous grid clustering is also presented. [Copyright &y& Elsevier]