In this paper, a high order finite difference conservative scheme is proposed to solve two-medium flows. Our scheme has four advantages: First, our scheme is conservative, which is important to ensure the numerical solution captures the main features properly. Second, our scheme directly applies the WENO interpolation method to the primitive variables so that it can maintain the equilibrium of velocity

In this paper, we construct a positivity-preserving high order accurate finite volume hybrid Hermite Weighted Essentially Non-oscillatory (HWENO) scheme for compressible Navier-Stokes equations, by incorporating a nonlinear flux and a positivity-preserving limiter. HWENO schemes have more compact stencils than WENO schemes but with higher computational cost due to the auxiliary variables. The hybrid

Liquid-gas flows that involve compressibility effects occur in many engineering contexts, and high-fidelity simulations can unlock further insights and developments. Introducing several numerical innovations, this work details a collocated, volume-of-fluid, finite volume flow solver that is robust, conservative, and capable of simulating flows with shocks, liquid-gas interfaces, and turbulence. A novel

In this paper, we introduce a second-order unstructured finite-volume method developed to solve a conservative level-set equation in two- and three-dimensional geometries. The characteristic function for the flow is defined at the center of each cell while the face-normal velocities are calculated at the mid-points of the corresponding cell faces. An interpolation method independent of cell shape and

Hardware acceleration consists of offloading computational work to devices such as graphics processing units (GPUs) to produce overall speed-up. Algorithms and numerical methods must be constructed to suit the available hardware in order to effectively produce speed-up. In this work a numerical method is presented which can effectively use hardware acceleration to simulate incompressible turbulent

The thermodynamically consistent Cahn-Hilliard-Extended-Darcy (CHED) model has been used to describe transient motion of a binary incompressible fluid flow in porous media. In this paper, we develop a series of linear, second-order, energy-dissipation-rate preserving numerical algorithms for the CHED model based on the energy quadratization strategy. We first extend the incompressible CHED model into

We propose in this paper the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm to solve parabolic equations with heterogeneous coefficients efficiently. This algorithm combines the advantages of multiscale methods that can deal with heterogeneity in the spatial domain effectively, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available

The second paper of this series presents two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element (DGSEM) discretizations of the compressible magneto-hydrodynamics (MHD) equations. Specifically, we use the resistive GLM-MHD equations, which include a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM). For the continuous entropy

Stochastic kriging (SK) offers an explicit way to characterize heterogeneous noise variance in stochastic computer simulations and has gained considerable traction recently as a surrogate model. Nevertheless, SK relies on tedious Monte Carlo (MC) method to estimate the intrinsic variance at each design input. For computationally expensive simulations, the substantial replication effort has essentially

Certain generalized coherent states, so-called Hagedorn wavepackets, have been used to numerically solve the standard Schrödinger equation. We extend this approach and its recent enhancements to the magnetic Schrödinger equation for a time-dependent, spatially homogeneous magnetic field. We explain why Hagedorn wavepackets are naturally compatible with the aforementioned physical system. In numerical

We present a numerical method for two-phase incompressible Navier–Stokes equation with jump discontinuities in the normal projection of the stress tensor and in the material properties. Although the proposed method is only first-order accurate, it does capture discontinuities sharply, not neglecting nor omitting any component of the jump condition. Discontinuities in velocity gradient and pressure

Finite-difference (FD) methods for the wave equation are flexible, robust and easy to implement. However, they in general suffer from numerical dispersion. FD methods based on accuracy give good dispersion at low frequencies, but waves tend to disperse for higher wavenumbers. Moreover, waves in higher dimensions also suffer from dispersion errors in all propagation angles. In this work, we give a unified

In extreme learning machines (ELM) the hidden-layer coefficients are randomly set and fixed, while the output-layer coefficients of the neural network are computed by a least squares method. The randomly-assigned coefficients in ELM are known to influence its performance and accuracy significantly. In this paper we present a modified batch intrinsic plasticity (modBIP) method for pre-training the random

The Test Particle Monte Carlo is a known method to solve the steady state Boltzmann particle transport equation in rarefied gas systems. A description of the Test Particle Monte-Carlo procedure is outlined and the accuracy of this method is investigated by analyzing its consistency to experimental data of steady state effusions in isothermal systems. Computational results present deviations from the

We survey a number of moment hierarchies and test their performances in computing one-dimensional shock structures. It is found that for high Mach numbers, the moment hierarchies are either computationally expensive or hard to converge, making these methods questionable for the simulation of highly nonequilibrium flows. By examining the convergence issue of Grad's moment methods, we propose a new moment

In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of

We introduce a class of Sparse, Physics-based, and partially Interpretable Neural Networks (SPINN) for solving ordinary and partial differential equations (PDEs). By reinterpreting a traditional meshless representation of solutions of PDEs we develop a class of sparse neural network architectures that are partially interpretable. The SPINN model we propose here serves as a seamless bridge between two

A general method is proposed to build exact artificial boundary conditions for the one-dimensional nonlocal Schrödinger equation. To this end, we first consider the spatial semi-discretization of the nonlocal equation, and then develop an accurate numerical method for computing the Green's function of the semi-discrete nonlocal Schrödinger equation. These Green's functions are next used to build the

Accurate numerical simulation of moving contact lines on complex boundaries with surface wettability effect remains a challenging problem. In this paper, we introduce a robust and accurate method to perform 3D contact line simulations on unstructured meshes with an imposed contact angle. The contact angle is imposed through a sub-grid scale curvature term and the contact line motion is enabled thanks

This article presents a new method to sample on manifolds, based on the Dirichlet distribution. The proposed strategy allows to completely respect the underlying manifold around which data are observed, and to do massive sampling with low computational effort. This can be very helpful, for instance, in neural networks training process, as well as in uncertainty analysis and stochastic optimization

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 extend are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for

We present a new approach to compute and analyze the dynamical electro-geometric properties of proteins undergoing conformational changes. The molecular trajectory is obtained from Markov state models, and the electrostatic potential is calculated using the continuum Poisson-Boltzmann equation. The numerical electric potential is constructed using a parallel sharp numerical solver implemented on adaptive

We establish a machine learning model for the prediction of the magnetization dynamics as function of the external field described by the Landau-Lifschitz-Gilbert equation, the partial differential equation of motion in micromagnetism. The model allows for fast and accurate determination of the response to an external field which is illustrated by a thin-film standard problem. The data-driven method

In this work we present an adaptive boundary-integral equation method for computing the electromagnetic response of wave interactions in hyperbolic metamaterials. The indefiniteness of the permittivity tensor gives rise to preferential wave radiation within the propagating cone for the hyperbolic media, and this induces sharp transition for the solution of the integral equation across the cone boundary

This work concerns the numerical approximation of a multicomponent compressible Euler system for a fluid mixture in multiple space dimensions on unstructured meshes with a high-order discontinuous Galerkin spectral element method (DGSEM). We first derive an entropy stable (ES) and robust (i.e., that preserves the positivity of the partial densities and internal energy) three-point finite volume scheme

Smoothed Particle Hydrodynamics (SPH) has intrinsic advantages over the Finite Volume (FV) method for predicting multiphase flows, since the need for fluid phase interface reconstructions is avoided due to the natural advection of those interfaces based on the Lagrangian description of the control volumes. However, permeable boundary conditions are less straightforward to handle with SPH, even though

This paper gives a derivation of the two-temperature Euler plasma system from the two-fluid MHD model. The two-temperature Euler plasma system is proved to be an asymptotic regime for weakly magnetized plasma of the two-fluid MHD model. Our procedure is more general, enabling us to show that assumptions in previous derivations in literature are straightforward consequences of our work. We then propose

This paper presents nonlinear iterative methods for the fundamental thermal radiative transfer (TRT) model defined by the time-dependent multifrequency radiative transfer (RT) equation and the material energy balance (MEB) equation. The iterative methods are based on the nonlinear projection approach and use multiple grids in photon frequency. They are formulated by the high-order RT equation on a

We present a robust, highly accurate, and efficient positivity- and boundedness-preserving diffuse interface method for the simulations of compressible gas-liquid two-phase flows with the five-equation model by Allaire et al. using high-order finite difference weighted compact nonlinear scheme (WCNS) in the explicit form. The equation of states of gas and liquid are given by the ideal gas and stiffened

A generalized approach to decompose the compressible Navier-Stokes equations into an equivalent set of coupled equations for flow and acoustics is introduced. As a significant extension to standard hydrodynamic/acoustic splitting methods, the approach provides the essential coupling terms, which account for the feedback from the acoustics to the flow. A unique simplified version of the split equation

In this work we propose a new, arbitrary order space-time finite element discretisation for Hamiltonian PDEs in multisymplectic formulation. We show that the new method which is obtained by using both continuous and discontinuous discretisations in space, admits a local and global conservation law of energy. We also show existence and uniqueness of solutions of the discrete equations. Further, we illustrate

We consider the generalized Benjamin-Ono (gBO) equation on the real line, ut+∂x(−Hux+1mum)=0,x∈R,m=2,3,4,5, and perform numerical study of its solutions. We first compute the ground state solution to −Q−HQ′+1mQm=0 via Petviashvili's iteration method. We then investigate the behavior of solutions in the Benjamin-Ono (m=2) equation for initial data with different decay rates and show decoupling of the

The development of interface-capturing methods (such as level-set, phase-field or volume of fluid (VOF) methods) for arbitrary 3D grids has further highlighted the need for more accurate and efficient interface reconstruction procedures. In this work, we propose a new method for the extraction of isosurfaces on arbitrary polyhedra that can be used with advantage for this purpose. The isosurface is

In this paper we use neural networks to learn governing equations from data. Specifically we reconstruct the right-hand side of a system of ODEs x˙(t)=f(t,x(t)) directly from observed uniformly time-sampled data using a neural network. In contrast with other neural network-based approaches to this problem, we add a Lipschitz regularization term to our loss function. In the synthetic examples we observed

This paper presents a positive and asymptotic preserving scheme for the nonlinear gray radiative transfer equations. The scheme is constructed by combining the filtered spherical harmonics (FPN) method for the discretization of angular variable and with the framework of the unified gas kinetic scheme (UGKS) for the spatial- and time-discretization. The constructed scheme is almost free of ray effects

In this paper, we propose a family of second and third order temporal integration methods for systems of stiff ordinary differential equations, and explore their application in solving the shallow water equations with friction. The new temporal discretization methods come from a combination of the traditional Runge-Kutta method (for non-stiff equation) and exponential Runge-Kutta method (for stiff

The exit time probability, which gives the likelihood that an initial condition leaves a prescribed region of the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems. Here we present an accurate and efficient numerical method for computing this probability for systems described by non-autonomous (time-dependent) stochastic

Dynamic mode decomposition (DMD) is a powerful data-driven technique for construction of reduced-order models of complex dynamical systems. Multiple numerical tests have demonstrated the accuracy and efficiency of DMD, but mostly for systems described by homogeneous partial differential equations (PDEs) with homogeneous boundary conditions. We propose an extended dynamic mode decomposition (xDMD) approach

Timely completion of design cycles for complex systems ranging from consumer electronics to hypersonic vehicles relies on rapid simulation-based prototyping. The latter typically involves high-dimensional spaces of possibly correlated control variables (CVs) and quantities of interest (QoIs) with non-Gaussian and possibly multimodal distributions. We develop a model-agnostic, moment-independent global

This paper deals with the development of efficient incomplete multidimensional Riemann solvers for hyperbolic systems. Departing from a four-waves model for the speeds of propagation arising at each vertex of the computational structured mesh, we present a general strategy for constructing genuinely multidimensional Riemann solvers, that can be applied for solving systems including source and coupling

An important class of multi-scale flow scenarios deals with an interplay between kinetic and continuum phenomena. While hybrid solvers provide a natural way to cope with these settings, two issues restrict their performance. Foremost, the inverse problem implied by estimating distributions has to be addressed, to provide boundary conditions for the kinetic solver. The next issue comes from defining

Compositional simulation for flow and transport in porous media is essential for modeling Enhanced Oil Recovery (EOR) processes. The Sequential Fully Implicit (SFI) scheme is considered as a more flexible and potentially more computationally efficient alternative to the commonly used Fully Implicit Method (FIM). SFI was proposed along with the multiscale method where the flow and transport problems

A deep-learning based approach is developed for efficient evaluation of thermophysical properties in numerical simulation of complex real-fluid flows. The work enables a significant improvement of computational efficiency by replacing direct calculation of the equation of state with a deep feedforward neural network with appropriate boundary information (DFNN-BC). The proposed method can be coupled

This paper presents a single-pressure-field two-fluid model with finite-volume discretization to solve the equations of motion of compressible multiphase flows. To capture the discontinuities caused by shock waves and fluid interfaces, we propose a generalized discontinuity sharpening technique that combines the conventional monotonic upstream scheme for conservation law (MUSCL) and tangent of hyperbola

The recently developed Deep Potential [Phys. Rev. Lett. 120 (2018) 143001 [27]] is a powerful method to represent general inter-atomic potentials using deep neural networks. The success of Deep Potential rests on the proper treatment of locality and symmetry properties of each component of the network. In this paper, we leverage its network structure to effectively represent the mapping from the atomic

Though the method-of-moments implementation of the electric-field integral equation plays an important role in computational electromagnetics, it provides many code-verification challenges due to the different sources of numerical error. In this paper, we provide an approach through which we can apply the method of manufactured solutions to isolate and verify the solution-discretization error. We accomplish

Sparse Identification of Nonlinear Dynamics (SINDy) is a method of system discovery that has been shown to successfully recover governing dynamical systems from data [6], [39]. Recently, several groups have independently discovered that the weak formulation provides orders of magnitude better robustness to noise. Here we extend our Weak SINDy (WSINDy) framework introduced in [28] to the setting of

Mesh-free methods such a smoothed particle hydrodynamics (SPH) have advantages over mesh-based methods for flow in complex domains but attaining consistent high-order accurate solutions with the conventional form of SPH has yet to be resolved. The high-order smoothing error dominates the SPH error until increasingly fine resolutions cause the low-order discretisation error to dominate; this is related

A sequential estimator based on the Ensemble Kalman Filter for Data Assimilation of fluid flows is presented in this research work. The main feature of this estimator is that the Kalman filter update, which relies on the determination of the Kalman gain, is performed exploiting the algorithmic features of the numerical solver employed as a model. More precisely, the multilevel resolution associated

The magnetotelluric (MT) method is an important geophysical electromagnetic induction exploration tool, with proper inversion routines, we can image the electrical conductivity structures of the Earth. A reliable inversion routine needs to iteratively call forward modeling solvers which solve the electromagnetic induction equations in the Earth. In this study, a novel hybrid Helmholtz-curl formulae

A novel approach to efficiently model 3-D laser plasma interactions at fluid scales is presented. This method, implemented in the IFRIIT propagation code developed at CELIA, relies on inverse ray tracing to compute laser fields at arbitrary locations in a plasma. This enables to describe the fields at high order in space compared to standard forward ray tracing approaches. In addition, inverse ray

A numerical method based on smoothed particle hydrodynamics with adaptive spatial resolution (SPH-ASR) was developed for simulating free surface flows. This method can reduce the computational demands while maintaining the numerical accuracy. In this method, the spatial resolution changes adaptively according to the distance to the free surface by numerical particle splitting and merging. The particles

The Note presents an alternative derivation and interpretation of the Germano identity and its error, showing that the Germano identity error directly estimates the residual of the LES equation (i.e., the misfit when evaluating the inexact equation for the exact solution) and therefore represents the source of errors in LES. The most prominent applications include a robust and easy to compute definition

The Cahn-Hilliard-Navier-Stokes (CHNS) equations represent the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Due to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation, the CHNS system is non-trivial to be solved numerically. Traditionally, a numerical extrapolation for the coupling terms is used. However, such brute-force

Using generative models from the machine learning literature to create artificial ensemble members for use within data assimilation schemes has been introduced in Grooms (2021) [1] as constructed analog ensemble optimal interpolation (cAnEnOI). Specifically, we study general and variational autoencoders for the machine learning component of this method, and combine the ideas of constructed analogs

A multiscale fractured-porous medium has several hierarchical levels of heterogeneity and consists of a connected network of fractures and of a periodic system of low permeable blocks, each of them also consisting of smaller fractures and smaller blocks. Repeating this procedure n times, we obtain a fractured-porous medium having n scales. The numerical simulation of a fluid flow in such a medium is

In this paper, a centred universal high-order finite volume method for solving hyperbolic balance laws is presented. The scheme belongs to the family of ADER methods where the Generalized Riemann Problems (GRP) is a building block. The solution to these problems is carried through an implicit Taylor series expansion, which allows the scheme to work very well for stiff source terms. The series expansion

A fully discrete finite element method, based on a new weak formulation and a new time-stepping scheme, is proposed for the surface diffusion flow of closed curves in the two-dimensional plane. It is proved that the proposed method can preserve two geometric structures simultaneously in the discrete level, i.e., the perimeter of the curve decreases in time while the area enclosed by the curve is conserved

In simulations, artificial boundaries need to be introduced to limit the size of computational domains and thereby lower computational cost. At these boundaries, flow variables need to be calculated in a way that will not induce any perturbation of the interior solution, which poses a great challenge in incompressible flows. In this paper, we demonstrate the potential of a new traction open boundary