Modal and nonmodal analyses of fluid flows provide fundamental insight into the early stages of transition to turbulence. Eigenvalues of the dynamical generator govern temporal growth or decay of individual modes, while singular values of the frequency response operator quantify the amplification of disturbances for linearly stable flows. In this paper, we develop well-conditioned ultraspherical and

A general, two-way coupled, point-particle formulation that accounts for the disturbance created by the dispersed particles in obtaining the undisturbed fluid flow field needed for accurate computation of the force closure models is presented. Specifically, equations for the disturbance field created by the presence of particles are first derived based on the inter-phase momentum coupling force in

Lorentz invariant structure-preserving algorithms possess reference-independent secular stability, which is vital for simulating relativistic multi-scale dynamical processes. The splitting method has been widely used to construct structure-preserving algorithms, but without exquisite considerations, it can easily break the Lorentz invariance of continuous systems. In this paper, we introduce a Lorentz

The purpose of this paper is to present a High-Order/Low-Order radiation-hydrodynamics method that is second-order accurate in both space and time and uses the Variable Eddington Factor (VEF) method to couple a high-order set of 1-D slab-geometry grey Sn radiation transport equations with a low-order set of radiation moment and hydrodynamics equations. The Sn equations are spatially discretized with

When the surface tension force acts on the multi-layer particles near the free surface, the integral of the previous surface delta function through the free surface is less than 1 because of the boundary truncation and kernel truncation, which leads to the problem that the numerical results cannot converge to the theoretical solution. Although a few studies have recognized this problem, it has not

In this paper, we propose a fast multipole method (FMM) for 3-D linearized Poisson–Boltzmann (PB) equation in layered electrolyte-dielectric media. We will extend our previous work on FMMs for Helmholtz and Laplace equations in layered media [1], [2] to the case of electrolyte and dielectric layered media for applications arising from biophysics and colloidal fluids, such as ion channel transport and

The numerical simulation of the equilibrium of the plasma in a tokamak as well as its self-consistent coupling with resistive diffusion should benefit from higher regularity of the approximation of the magnetic flux map. In this work, we propose a finite element approach on a triangular mesh of the poloidal section, that couples piece-wise linear finite elements in a region that does not contain the

The continuum description of active particle systems is an efficient instrument to analyze a finite size particle dynamics in the limit of a large number of particles. However, it is often the case that such equations appear as nonlinear integro-differential equations and purely analytical treatment becomes quite limited. We propose a general framework of finite volume methods (FVMs) to numerically

The direct numerical simulation (DNS) of compressible isotropic turbulence up to the supersonic regime with Mat=1.2 has been investigated by the high-order gas-kinetic scheme (HGKS) [Computers & Fluids, 192 (2019) 104273]. In this study, the DNS on a much higher initial turbulent Mach number up to Mat=2.0 is performed by HGKS, which confirms the super robustness of HGKS. The coarse-graining analysis

In this paper we analyze the noise in macro-particle methods used in plasma physics and fluid dynamics, leading to approaches for minimizing the total error, focusing on electrostatic models in one dimension. We begin by describing kernel density estimation for continuous values of the spatial variable x, expressing the kernel in a form in which its shape and width are represented separately. The covariance

A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. The MBP plays a crucial role in understanding the physical meaning and the wellposedness of the mathematical model. Investigation on numerical algorithms with preservation

The scope of this paper is to demonstrate the viability of unstructured anisotropic mesh adaptation for commercial aircraft drag and high-lift prediction studies. The main achievement of this work is to demonstrate that mesh-independent certified numerical solutions can be achievable thanks to anisotropic mesh adaptation and that it is possible to run high-fidelity CFD on unstructured adapted meshes

We propose a dynamical low-rank method to reduce the computational complexity for solving the multi-scale multi-dimensional linear transport equation. The method is based on a macro-micro decomposition of the equation; the low-rank approximation is only used for the micro part of the solution. The time and spatial discretizations are done properly so that the overall scheme is second-order accurate

In this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compared with a traditional nonlinear POD (NPOD) model by

We describe a new algorithm, vegas+, for adaptive multidimensional Monte Carlo integration. The new algorithm adds a second adaptive strategy, adaptive stratified sampling, to the adaptive importance sampling that is the basis for its widely used predecessor vegas. Both vegas and vegas+ are effective for integrands with large peaks, but vegas+ can be much more effective for integrands with multiple

This paper proposes a multiscale method for solving the numerical solution of mean field games which accelerates the convergence and addresses the problem of determining the initial guess. Starting from an approximate solution at the coarsest level, the method constructs approximations on successively finer grids via alternating sweeping, which not only allows for the use of classical time marching

Problems involving a transition between dense and dilute dispersed phase regimes often require hybridization of Eulerian-Eulerian (EE) and Eulerian-Lagrangian (EL) methods for accurate and efficient computational modeling. A hybrid EE-EL formulation is developed in this work from first principles, which asymptotes to well-established EE and EL methods in limiting conditions. A smooth and dynamic transition

In this paper, we study temporal splitting algorithms for multiscale problems. The exact fine-grid spatial problems typically require some reduction in degrees of freedom. Multiscale algorithms are designed to represent the fine-scale details on a coarse grid and, thus, reduce the problems' size. When solving time-dependent problems, one can take advantage of the multiscale decomposition of the solution

High-order numerical methods for solving elliptic equations over arbitrary domains typically require specialized machinery, such as high-quality conforming grids for finite elements method, and quadrature rules for boundary integral methods. These tools make it difficult to apply these techniques to higher dimensions. In contrast, fixed Cartesian grid methods, such as the immersed boundary (IB) method

Extended Lagrangian molecular dynamics (XLMD) is a general method for performing molecular dynamics simulations using quantum and classical many-body potentials. Recently several new XLMD schemes have been proposed and tested on several classes of many-body polarization models such as induced dipoles or Drude charges, by creating an auxiliary set of these same degrees of freedom that are reversibly

Applications for kinetic equations such as optimal design and inverse problems often involve finding unknown parameters through gradient-based optimization algorithms. Based on the adjoint-state method, we derive two different frameworks for approximating the gradient of an objective functional constrained by the nonlinear Boltzmann equation. While the forward problem can be solved by the DSMC method

The Parallel Block Jacobi (PBJ) [1], [2] scheme is an iterative method with application in neutron transport solution which requires asynchronous sub-domain interfaces, where asynchronous here means flux continuity is not imposed across sub-domain interfaces. In this work, we present the development, analysis, and testing of hybrid methods which utilize both PBJ-type methods and synchronous sweep-based

We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method formulated by introducing a modified adjoint problem for the test function and by performing the integration of PDE over a space-time region partitioned by time-dependent linear functions approximating characteristics. The error incurred in characteristics approximation in the modified adjoint problem can then be taken into

We study in this paper three variants of the high-order Discontinuous Galerkin (DG) method with Runge-Kutta (RK) time integration for the induction equation, analysing their ability to preserve the divergence-free constraint of the magnetic field. To quantify divergence errors, we use a norm based on both a surface term, measuring global divergence errors, and a volume term, measuring local divergence

In this study, a novel two-dimensional finite-volume method on unstructured triangular meshes for two-layer shallow water flows is developed. First, the one-dimensional relaxation approach developed in Abgrall and Karni (2009) [1] is extended to the currently studied two-dimensional two-layer shallow water equations to build an unconditionally hyperbolic system. Next, a two-dimensional finite-volume

In inverse problems, redatuming data consists in virtually moving the sensors from the original acquisition location to an arbitrary position. This is an essential tool for target oriented inversion. An exact redatuming method which has the peculiarity to be robust with respect to noise is proposed. Our iterative method is based on the Time Reversal Absorbing Conditions (TRAC) approach [1] and avoids

Numerical solution of inverse problem for 2D acoustic system of conservation laws by gradient type method requires storage of O(N3) elements which is crucial on large grids with O(N) points in single dimension. In this article we present an approach to save twice memory on the stage of adjoint problem and gradient calculation and compare it with usual approach in memory and CPU time cost. Numerical

For high frequency Helmholtz equations with point-source conditions, geometrical optics provides an efficient way to compute the solutions without ‘pollution effect’ that is usually unavoidable in finite difference and finite element methods. However, geometrical optics generally can only provide locally valid approximations for the wavefield near the primary source. In order to obtain globally valid

A parallel computational method for simulating fluid–structure interaction problems involving large, geometrically nonlinear deformations of thin shell structures is presented and validated. A compressible Navier-Stokes solver utilizing a higher-order finite difference immersed boundary method is coupled with a geometrically nonlinear computational structural dynamics solver employing the mixed interpolation

This paper presents a simple and highly accurate method for capturing sharp interfaces moving in divergence-free velocity fields using the high-order Flux Reconstruction approach on unstructured grids. A well-known limitation of high-order methods is their susceptibility to the Gibbs phenomenon; the appearance of spurious oscillations in the vicinity of discontinuities and steep gradients makes it

We develop a framework and numerical method for controlling the full space-time tube of a geometrically driven flow. We consider an optimal control problem for the mean curvature flow of a curve or surface with a volume constraint, where the control parameter acts as a forcing term in the motion law. The control of the trajectory of the flow is achieved by minimizing an appropriate tracking-type cost

We develop a numerical scheme for Poisson-Boltzmann-Nernst-Planck (PBNP) model. We adopt Gummel's method to treat the nonlinearity of PBNP where Poisson-Bolzmann equation and Nernst-Plank equation are iteratively solved, and then the idea of discontinuous bubble (DB) to solve the Poisson-Bolzmann equation is exploited [6]. First, we regularize the solution of Poisson-Bolzmann equation to remove the

The ghost-cell immersed boundary methods (IBMs) are widely used to implement boundary conditions on non-body fitted grids. It has been previously shown that they have two major drawbacks. Firstly, these methods tend to have a maximum stencil size larger than 1, yielding non-band matrices. Secondly, for a maximum stencil size of 2 the ghost-cell linear IBM [1] have a first-order convergence rate for

Next-generation high-power laser systems that can be focused to ultra-high intensities exceeding 1023 W/cm2 are enabling new physics regimes and applications. The physics of how these lasers interact with matter is highly nonlinear, relativistic, and can involve lowest-order quantum effects. The current tool of choice for modeling these interactions is the particle-in-cell (PIC) method. In the presence

In this paper, we consider the quasi-gas-dynamic (QGD) model in a multiscale environment. The model equations can be regarded as a hyperbolic regularization and are derived from kinetic equations. So far, the research on QGD models has been focused on problems with constant coefficients. In this paper, we investigate the QGD model in multiscale media, which can be used in porous media applications

In this paper, we present a novel cell-centered indirect Arbitrary-Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme on moving unstructured triangular meshes with mesh topological adaptability, aiming to deal with the strong distortions and large deformation flow problems. The scheme combines the explicit time marching Lagrangian DG methodology with the adaptive mesh topology optimization

In this paper, we consider numerical approximations for solving the hydrodynamically coupled three components Cahn-Hilliard phase-field model. By combining the Invariant Energy Quadratization approach with the stabilization technique, and the projection method for the Navier-Stokes equations, we obtain a linear, second-order, and unconditionally energy stable time marching scheme. We present rigorous

The varying-mass Schrödinger equation (VMSE) has been successfully applied to model electronic properties of semiconductor hetero-structures, for example, quantum dots and quantum wells. In this paper, we consider VMSE with small random heterogeneities, and derive a radiative transfer equation as its asymptotic limit. The main tool is to systematically apply the Wigner transform in the classical regime

In this paper a fully Eulerian solver for the study of multiphase flows for simulating the propagation of surface gravity waves over submerged bodies is presented. We solve the incompressible Navier–Stokes equations coupled with the volume of fluid technique for the modeling of the liquid phases with the inteface, an immersed body method for the solid objects and an iterative strong–coupling procedure

Physical and numerical limitations of stationary Vlasov-Poisson solvers based on backward Liouville methods are investigated with five solvers that combine different meshes, numerical integrators, and electric field interpolation schemes. Since some of the limitations arise when moving from an integrable to a non-integrable configuration, an elliptical Langmuir probe immersed in a Maxwellian plasma

A multipole-accelerated 3D boundary-integral algorithm is developed to model pressure-driven flow of a highly-concentrated emulsion of many deformable drops through a periodic channel with tight constrictions. The drops are monodisperse, free from surfactant, and the Reynolds number is small for Stokes equations to apply. The channel surface consists of four solid panels: front and back parallel to

We present a general numerical approach for learning unknown dynamical systems using deep neural networks (DNNs). Our method is built upon recent studies that identified residual network (ResNet) as an effective neural network learning structure. In this paper, we present a generalized ResNet framework and broadly define “residue” as the discrepancy between observation data and prediction made by another

Gaseous flows show a diverse set of behaviors on different characteristic scales. Given the coarse-grained modeling in theories of fluids, considerable uncertainties may exist between the flow-field solutions and the real physics. To study the emergence, propagation and evolution of uncertainties from molecular to hydrodynamic level poses great opportunities and challenges to develop both sound theories

The compressible Euler equations coupled with the gravitational source terms admit a hydrostatic equilibrium state where the gradients of the flux terms can be exactly balanced by those in the source terms. This property of exact preservation of the equilibrium is highly desirable at the discrete level when they are numerically solved. In this study, we design the simple high order well-balanced finite

We develop a multirate timestepper for semi-implicit solutions of the unsteady incompressible Navier-Stokes equations (INSE) based on a recently-developed multidomain spectral element method (SEM) [1]. For incompressible flows, multirate timestepping (MTS) is particularly challenging because of the tight coupling implied by the incompressibility constraint, which manifests as an elliptic subproblem

We present a new family of high-order shock-capturing finite difference numerical methods for systems of conservation laws. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered (2p+1)-point stencils, where p may take values in {1,2,…,P} according to a new family of smoothness indicators in the stencils. The methods are based on a combination of a robust first order

We propose an efficient method to reinitialize a level set function to a signed distance function by solving an elliptic problem using the finite element method. The original zero level set interface is preserved by means of applying modified boundary conditions to a surrogate/approximate interface weakly with a penalty method. Narrow band technique is adopted to reinforce the robustness of the proposed

We propose a new class of Bayesian neural networks (BNNs) that can be trained using noisy data of variable fidelity, and we apply them to learn function approximations as well as to solve inverse problems based on partial differential equations (PDEs). These multi-fidelity BNNs consist of three neural networks: The first is a fully connected neural network, which is trained following the maximum a

An alternating directional implicit (ADI)-Yee's scheme is developed for Maxwell's equations with discontinuous material coefficients along one or several interfaces. In order to use Yee's scheme with the presence of discontinuities, some intermediate quantities along the interface are introduced. The intermediate quantities are from the solutions and their derivatives on the interface and should satisfy

The goal of this work is to train a neural network which approximates solutions to the Navier-Stokes equations across a region of parameter space, in which the parameters define physical properties such as domain shape and boundary conditions. The contributions of this work are threefold: 1. To demonstrate that neural networks can be efficient aggregators of whole families of parameteric solutions

In this paper, we propose a procedure combining Lattice-Boltzmann and finite element simulations to model the effects of capillary pressure in porous microstructures. Starting from an explicit geometry of the microstructure, the Lattice-Boltzmann method is used to simulate the condensation from vapor phase to liquid and predict the geometry of capillary liquid films and liquid phases for arbitrary

In this paper, a domain decomposition technique in the finite volume framework is presented to propagate small amplitude acoustic and entropy waves in a linearized Euler region and simulate the interaction of these waves with an initially steady normal shock in a nonlinear region. An overset method is used to two-way couple the linear and nonlinear regions that overlap each other. Linearized solvers

We propose an accelerated density matrix purification scheme with error control. The method makes use of the scale-and-fold acceleration technique and screening of submatrix products in the block-sparse matrix-matrix multiplies to reduce the computational cost. An error bound and a parameter sweep are combined to select a threshold value for the screening, such that the error can be controlled. We

The leapfrog integrator is routinely used within the Hamiltonian Monte Carlo method and its variants. We give strong numerical evidence that alternative, easy to implement algorithms yield fewer rejections with a given computational effort. When the dimensionality of the target distribution is high, the number of accepted proposals may be multiplied by a factor of three or more. This increase in the

Tracking methods are used in particle transport Monte Carlo simulations to simulate the particle movements in models with multiple materials. However, as Monte Carlo simulations are becoming increasingly sophisticated, the most widely used tracking method, ray-tracing, exposes serious disadvantages in dealing with continuously varying materials and needs to be improved. Delta-tracking, the most common

The discontinuous Galerkin domain decomposition (DGDD) method couples subdomains of high-fidelity polynomial approximation to regions of low-dimensional resolution for the numerical solution of systems of conservation laws. In the low-fidelity regions, the solution is approximated by empirical modes constructed by Proper Orthogonal Decomposition and a reduced-order model is used to predict the solution

Rational exponential integrators (REXI) are a class of numerical methods that are well suited for the time integration of linear partial differential equations with imaginary eigenvalues. Since these methods can be parallelized in time (in addition to the spatial parallelization that is commonly performed) they are well suited to exploit modern high performance computing systems. In this paper, we

The paper is concerned with the three-dimensional electromagnetic scattering from a large open rectangular cavity that is embedded in a perfectly electrically conducting infinite ground plane. By introducing a transparent boundary condition, the scattering problem is formulated into a boundary value problem in the bounded cavity. Based on the Fourier expansions of the electric field, the Maxwell equation

In this paper, based on the Scalar Auxiliary Variable (SAV) approach [44], [45] and a newly proposed Lagrange multiplier (LagM) approach [22], [21] originally constructed for gradient flows, we propose two linear implicit pseudo-spectral schemes for simulating the dynamics of general nonlinear Schrödinger/Gross-Pitaevskii equations. Both schemes are of spectral/second-order accuracy in spatial/temporal

It is crucial to understand the degradation of implosion performance caused by three-dimensional physics and the implementation of 3-D spherical mesh is challenging. In this paper a mesh generation method based on the cube spherical projection is proposed for 3D implosion problem in Inertial Confinement Fusion (ICF). This kind of mesh has the advantage of good quality, convenient adding radiation source