We present a variational assimilation technique (4D-Var) to reconstruct time resolved incompressible turbulent flows from measurements on two orthogonal 2D planes. The proposed technique incorporates an error term associated to the flow dynamics. It is therefore a compromise between a strong constraint assimilation procedure (for which the dynamical model is assumed to be perfectly known) and a weak

In this paper, we present a new implicit Monte-Carlo scheme for photonics. The new solver combines the benefits of both the IMC solver of Fleck & Cummings and the SMC solver of Ahrens & Larsen. It is implicit hence allows taking affordable time steps (as IMC) and has no teleportation error (as SMC). The paper also provides some original analysis of existing schemes (IMC, tilted IMC, SMC), especially

The eigensystem of the Pulliam-Chaussee diagonalized form of the approximate-factorization algorithm for the three-dimensional Euler and Navier-Stokes equations is revisited to remove an apparent dimensional inconsistency. The original set of eigenvectors in curvilinear coordinates were derived systematically and has been widely used and referenced. Although mathematically correct, the original eigenvectors

The particle-in-cell (PIC) method is widely used to model the self-consistent interaction between discrete particles and electromagnetic fields. It has been successfully applied to problems across plasma physics including plasma based acceleration, inertial confinement fusion, magnetically confined fusion, space physics, astrophysics, high energy density plasmas. In many cases the physics involves

In this article, we propose a novel method for sampling potential functions based on noisy observation data of a finite number of observables in quantum canonical ensembles, which leads to the accurate sampling of a wide class of test observables. The method is based on the Bayesian inversion framework, which provides a platform for analyzing the posterior distribution and naturally leads to an efficient

A hybrid method is developed to solve the interaction problem of wave with a three dimensional floating structure in a polynya. The linearized velocity potential theory is used for fluid flow, and the thin elastic plate model is adopted for the infinitely extended ice sheet. Because of sudden change of the upper boundary of the computational domain, namely from the ice sheet to the free surface, the

We propose a novel approach to approximate numerically shock waves. The method combines the unstructured shock-fitting approach developed in the last decade by some of the authors, with ideas coming from embedded boundary techniques. The numerical method obtained allows avoiding the re-meshing phase required by the unstructured fitting method, while guaranteeing accuracy properties very close to those

This paper presents a convective time-domain equivalent-source method for determining the scattered acoustic pressure field in a uniform moving medium. The proposed method is based on the solution of the time-domain convective Ffowcs Williams–Hawkings (FW–H) equation while the strengths of equivalent sources are determined by the required pressure gradient boundary condition on the scattering surface

We construct a finite element discretization and time-stepping scheme for the incompressible Euler equations with variable density that exactly preserves total mass, total squared density, total energy, and pointwise incompressibility. The method uses Raviart-Thomas or Brezzi-Douglas-Marini finite elements to approximate the velocity and discontinuous polynomials to approximate the density and pressure

We present an arbitrarily high-order, conditionally stable, partitioned spectral deferred correction (SDC) method for solving multiphysics problems using a sequence of pre-existing single-physics solvers. This method extends the work in [1], [2], which used implicit-explicit Runge-Kutta methods (IMEX) to build high-order, partitioned multiphysics solvers. We consider a generic multiphysics problem

An accurate, efficient, and conceptually simple method is developed to estimate distributions of maxima of solutions of stochastic equations, i.e., ordinary or partial differential equations with random entries. The method is data-based. It constructs importance sampling (IS) or biasing measures from samples of surrogates of full model solutions of stochastic equations and uses these measures and mixtures

In this work, we propose a flux solver that relies on a double distribution function (DDF) based discrete gas-kinetic scheme (DGKS) for incompressible and compressible viscous flows. By decomposing the phase space of the Maxwellian distribution function, the double distribution function (DDF) model is established to remove the phase energy variables and develop more compact formulations. It utilizes

Transport of electrolytic solutions under influence of electric fields occurs in phenomena ranging from biology to geophysics. Here, we present a continuum model for single-phase electrohydrodynamic flow, which can be derived from fundamental thermodynamic principles. This results in a generalized Navier–Stokes–Poisson–Nernst–Planck system, where fluid properties such as density and permittivity depend

The WT scheme, a piecewise polynomial force interpolation scheme with time-step dependency, is proposed in this paper for relativistic particle-in-cell (PIC) simulations. The WT scheme removes the lowest order numerical Cherenkov instability (NCI) growth rate for arbitrary time steps allowed by the Courant condition. While NCI from higher order resonances is still present, the numerical tests show

A high-order accurate scheme for solving the time-domain dispersive Maxwell's equations and material interfaces is described. Maxwell's equations are solved in second-order form for the electric field. A generalized dispersive material (GDM) model is used to represent a general class of linear dispersive materials and this model is implemented in the time-domain with the auxiliary differential equation

Numerical Cherenkov radiation is a well-known problem of FDTD PIC codes caused by numerical dispersion errors creating spurious radiation around accelerated particles. It is usually dealt with by complex numerical schemes and low-pass filters. In this study we look at a simple way to circumvent this problem for wakefield accelerators simulations by only modifying the field interpolation algorithm.

This paper focuses on the fast evaluation of the matrix-vector multiplication (matvec) g=Kf for K∈CN×N, which is the discretization of a multidimensional oscillatory integral transform g(x)=∫K(x,ξ)f(ξ)dξ with a kernel function K(x,ξ)=e2πiΦ(x,ξ), where Φ(x,ξ) is a piecewise smooth phase function with x and ξ in Rd for d=2 or 3. A new framework is introduced to compute Kf with O(Nlog⁡(N)) time and memories

The paper proposes a density gradient based approach to topology optimization under design-dependent boundary loading. Since there is no explicit boundary representation in density-based topology optimization, the design-dependent boundary loads are implicitly imposed through a domain integration of spatial gradient of the density field. The density gradient based loading is formally derived and justified

In this paper, a face-averaged nodal-gradient approach is proposed as an efficient gradient method for a second-order cell-centered finite-volume discretization on triangular grids. The gradients needed in the linear reconstruction are computed in two steps: (1) compute gradients at nodes from solutions stored at cells, and (2) compute the gradient at each face by averaging the nodal gradients over

We develop a well-balanced central-upwind scheme for rotating shallow water model with horizontal temperature and/or density gradients—the thermal rotating shallow water (TRSW). The scheme is designed using the flux globalization approach: first, the source terms are incorporated into the fluxes, which results in a hyperbolic system with global fluxes; second, we apply the Riemann-problem-solver-free

In this work, we present a novel nonlocal nonlinear coarse grid approximation using a machine learning algorithm. We consider unsaturated and two-phase flow problems in heterogeneous and fractured porous media, where mathematical models are formulated as general multicontinuum models. We construct a fine grid approximation using the finite volume method and embedded discrete fracture model. Macroscopic

This paper presents a novel CFD-driven machine learning framework to develop Reynolds-averaged Navier-Stokes (RANS) models. The CFD-driven training is an extension of the gene expression programming method Weatheritt and Sandberg (2016) [8], but crucially the fitness of candidate models is now evaluated by running RANS calculations in an integrated way, rather than using an algebraic function. Unlike

This paper investigates the construction and optimization of multidimensional summation-by-parts (SBP) operators. The optimization of free parameters in the cubature rule, the interpolation/extrapolation operator R, and in the skew-symmetric matrix Sξ, is investigated and a methodology to optimize the operators is presented. It is demonstrated that increasing the degree of the cubature rule from 2p−1

Pseudospectral numerical schemes for solving the Dirac equation in general static curved space are derived using a new pseudodifferential representation of the Dirac equation along with a simple Fourier-basis technique. Owing to the presence of non-constant coefficients in the curved space Dirac equation, convolution products usually appear when the Fourier transform is performed. To circumvent this

This work extends the concepts of algebraic flux correction and convex limiting to continuous high-order Bernstein finite element discretizations of scalar hyperbolic problems. Using an array of adjustable diffusive fluxes, the standard Galerkin approximation is transformed into a nonlinear high-resolution scheme which has the compact sparsity pattern of the piecewise-linear or multilinear subcell

Solving general high-dimensional partial differential equations (PDE) is a long-standing challenge in numerical mathematics. In this paper, we propose a novel approach to solve high-dimensional linear and nonlinear PDEs defined on arbitrary domains by leveraging their weak formulations. We convert the problem of finding the weak solution of PDEs into an operator norm minimization problem induced from

In this article, stochastic conformal schemes of damped stochastic Klein-Gordon equation with additive noise are studied. It is shown that this equation possesses stochastic conformal multi-symplectic conservation law. Under appropriate boundary conditions, global momentum evolution law and global energy evolution law are proposed. We chiefly develop stochastic conformal Preissman scheme, stochastic

Recently, several methods have been proposed to simulate incompressible fluid flows using an artificial pressure evolution equation, avoiding the resolution of a Poisson equation. These methods can be seen as various levels of approximation of the compressible Navier–Stokes equation in the low Mach number limit. We study the simulation of incompressible wall-bounded flows using several artificial pressure

A modal discontinuous Galerkin method was developed for computing compressible rarefied gaseous flows interacting with rigid particles and granular medium. In contrast to previous particle-based models that were developed to handle rarefied flows or solid phase particles, the present computational method employs full continuum-based models. This work is one of the first attempts to apply the modal

In this article a domain decomposition method is proposed to solve a time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition and defective boundary condition. Robin boundary conditions between the Navier-Stokes domain and Darcy domain are constructed by directly re-organizing the terms in the three interface conditions, including the Beavers-Joseph condition. In order to avoid

Magnetic resonance electrical impedance tomography (MREIT) and current density imaging (MRCDI) are emerging as new methods to non-invasively assess the electric conductivity of and current density distributions within biological tissue in vivo. The accurate and fast computation of magnetic fields caused by low frequency electrical currents is a central component of the development, evaluation and application

We present a practical finite difference scheme for the incompressible Navier–Stokes equation on curved surfaces in three-dimensional space. In the proposed method, the curved surface is embedded in a narrow band domain and the governing equation is extended to the narrow band domain. We use the standard seven-point stencil for the Laplace operator instead of a discrete Laplacian–Beltrami operator

The Wang-Landau (WL) algorithm is a recently developed stochastic algorithm computing densities of states of a physical system, and also performing numerical integration in high dimensional spaces. Since its inception, it has been used on a variety of (bio-)physical systems, and in selected cases, its convergence has been proved. The convergence speed of the algorithm is tightly tied to the connectivity

We suggest in this paper a new stochastic simulation algorithm for solving narrow escape problems governed by drift-diffusion-reaction equations of high dimension. The developed method drastically improves the efficiency of the diffusion trajectory tracking algorithm by introducing an artificial drift directed to the target position. The method is especially appropriate to solve narrow escape problems

It is known that the Method of Moments is less accurate close to the wall where non-equilibrium effects are strong. We therefore propose a hybrid algorithm that combines the discrete velocity method with the Method of Moments to accurately simulate rarefied gas flows in the transition regime. A discrete velocity approach, combined with Maxwell's wall boundary condition, is employed in the near-wall

Quantifying uncertainty in predictive simulations for real-world problems is of paramount importance - and far from trivial, mainly due to the large number of stochastic parameters and significant computational requirements. Adaptive sparse grid approximations are an established approach to overcome these challenges. However, standard adaptivity is based on global information, thus properties such

Standard solvers for the variable coefficient Helmholtz equation in two spatial dimensions have running times which grow at least quadratically with the wavenumber k. Here, we describe a solver which applies only when the scattering potential is radially symmetric but whose running time is O(klog⁡(k)) in typical cases. We also present the results of numerical experiments demonstrating the properties

Scientific computations or measurements may result in huge volumes of high-dimensional data, for instance 1020 or 100300 elements. Often these can be thought of representing a real-valued function on a high-dimensional domain. In this and also in most other cases the data can be conceptually arranged in the format of a tensor of high degree, and stored in some truncated or lossy compressed format.

We discuss a numerical approach for the simulation of a hydrodynamic model for immiscible incompressible two-phase flow in heterogeneous porous media to deal with effects of dynamic capillary pressure and gravity. The governing system of equations consists of a nonlinear pseudo-parabolic Buckley-Leverett saturation transport model coupled with an elliptic pressure-velocity problem. Upon certain manipulation

We present and analyze a weak Galerkin finite element method for solving the transport-reaction equation in d space dimensions. This method is highly flexible by allowing the use of discontinuous finite element on general meshes consisting of arbitrary polygon/polyhedra. We derive the L2-error estimate of O(hk+12)-order for the discrete solution when the kth-order polynomials are used for k≥0. Moreover

Closure modeling based on the Mori-Zwanzig formalism has proven effective to improve the stability and accuracy of projection-based model order reduction. However, closure models are often expensive and infeasible for complex nonlinear systems. Towards efficient model reduction of general problems, this paper presents a recurrent neural network (RNN) closure of parametric POD-Galerkin reduced-order

The threshold dynamics algorithm of Merriman, Bence, and Osher is only first order accurate in the two-phase setting. Its accuracy degrades further to half order in the multi-phase setting, a shortcoming it has in common with other related, more recent algorithms such as the equal surface tension version of the Voronoi implicit interface method. As a first, rigorous step in addressing this shortcoming

A methodology for modelling cavitating flows using a high-order Adaptive Mesh Refinement (AMR) approach based on the Discontinuous Galerkin method (DG) is presented. The AMR implementation used features on-the-fly adaptive mesh refinement for unstructured hybrid meshes. The specific implementation has been developed for the resolution of complex multi-scale phenomena where high accuracy p-adaptive

Pseudodifferential mode parabolic equations in curvilinear coordinates are derived from the horizontal refraction equations. An energy-conserving split-step Padé method for their numerical solution is developed. Several examples are considered including the propagation of sound in a perfect wedge, a whispering gallery formation near circular isobath, and the breaking of the whispering gallery waveguide

Anisotropic mesh quality measures and adaptation are studied for convex polygonal meshes. Three sets of alignment and equidistribution measures are developed, based on least squares fitting, generalized barycentric mappings, and the singular value decomposition, respectively. Numerical tests show that all three sets of mesh quality measures provide good measurements for the quality of polygonal meshes

We introduce solution dependent finite difference stencils whose coefficients adapt to the current numerical solution by minimizing the truncation error in the least squares sense. The resulting scheme has the resolution capacity of dispersion relation preserving difference stencils in under-resolved domains, together with the high order convergence rate of conventional central difference methods in

In this work, a stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm

Many large scale problems in computational fluid dynamics such as uncertainty quantification, Bayesian inversion, data assimilation and PDE constrained optimization are considered very challenging computationally as they require a large number of expensive (forward) numerical solutions of the corresponding PDEs. We propose a machine learning algorithm, based on deep artificial neural networks, that

We develop an entropy–stable two–phase incompressible Navier–Stokes/Cahn–Hilliard discontinuous Galerkin (DG) flow solver method. The model poses the Cahn–Hilliard equation as the phase field method, a skew–symmetric form of the momentum equation, and an artificial compressibility method to compute the pressure. We design the model so that it satisfies an entropy law, including free– and no–slip wall

We present a fast, high-order accurate and adaptive boundary integral scheme for solving the Stokes equations in complex—possibly nonsmooth—geometries in two dimensions. We apply the panel-based quadratures of Helsing and coworkers to evaluate to high accuracy the weakly-singular, hyper-singular, and super-singular integrals arising in the Nyström discretization, and also the near-singular integrals

We present an efficient method to solve the Poisson equation in embedded boundary (EB) domains. The original problem is divided into an inhomogeneous problem without the effects of EB and a homogeneous problem that imposes the effects of EB. The inhomogeneous problem is efficiently solved through a geometric multi-grid (GMG) solver and the homogenous problem is solved through a boundary element method

The Mittag-Leffler function (MLF) is fundamental to solving fractional calculus problems. An exponential sum approximation for the single-parameter MLF with negative input is proposed. Analysis shows that the approximation, which is based on the Gauss-Legendre quadrature, converges uniformly for all non-positive input. The application to modelling of wave propagation in viscoelastic material with the

In this paper, a unified approach is introduced to implement high order central difference schemes for solving Poisson's equation via the fast Fourier transform (FFT). Popular high order fast Poisson solvers in the literature include compact finite differences and spectral methods. However, FFT-based high order central difference schemes have never been developed for Poisson problems, because with

A Discontinuous Galerkin (DG) method for the solution of the shallow-water equations (SWE) on arbitrary 2-dimensional (2D) manifolds is presented. To this purpose the SWE are formulated in covariant form using tensor notation. This allows to correctly transform the numerical fluxes between the local coordinate systems of any two neighbouring grid cells. In particular, the covariant form of the numerical

We develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner. The use of a covariant basis decomposition leads to a rotationally invariant scheme that outperforms a Cartesian basis decomposition on rotated grids. The discretization is based on high order Summation-By-Parts (SBP) operators and preserves both symmetry

A nonlinear optimal smoother and an associated optimal strategy of sampling hidden model trajectories are developed for a rich class of complex nonlinear turbulent dynamical systems with partial and noisy observations. Despite the strong nonlinearity and the significant non-Gaussian characteristics in the underlying systems, both the optimal smoother estimates and the sampled trajectories can be solved

A mechanism is described to symmetrize the ultraspherical spectral method for self-adjoint problems. The resulting discretizations are symmetric and banded. An algorithm is presented for an adaptive spectral decomposition of self-adjoint operators. Several applications are explored to demonstrate the properties of the symmetrizer and the adaptive spectral decomposition.

A novel framework for resolving discontinuous solutions of conservation laws, e.g., contact lines, shock waves, and interfaces, using implicit tracking and a high-order discontinuous Galerkin (DG) discretization was introduced in [39]. Central to the framework is an optimization problem whose solution is a discontinuity-aligned mesh and the corresponding high-order approximation to the flow that does

This article presents explicit exponential integrators for stochastic Maxwell's equations driven by both multiplicative and additive noises. By utilizing the regularity estimate of the mild solution, we first prove that the strong order of the numerical approximation is 12 for general multiplicative noise. Combining a proper decomposition with the stochastic Fubini's theorem, the strong order of the