In the present work, we propose a consistent and conservative model for multiphase and multicomponent incompressible flows, where there can be arbitrary numbers of phases and components. Each phase has a background fluid called the pure phase, each pair of phases is immiscible, and components are dissolvable in some specific phases. The model is developed based on the multiphase Phase-Field model including

In this work, we consider discretization of the linearized shallow water equations on a reduced latitude-longitude grid with an analogue of Arakawa C-type variables staggering. The resulting schemes are based on the use of longitudinal interpolation procedures and can be of arbitrary order of accuracy. We also present the analysis of conservation and wave propagation properties of these schemes for

We present a simple and effective method for representing periodic functions and enforcing exactly the periodic boundary conditions for solving differential equations with deep neural networks (DNN). The method stems from some simple properties about function compositions involving periodic functions. It essentially composes a DNN-represented arbitrary function with a set of independent periodic functions

This paper proposes a novel fixed inducing points online Bayesian calibration (FIPO-BC) algorithm to efficiently learn the model parameters using a benchmark database. The standard Bayesian calibration (STD-BC) algorithm provides a statistical method to calibrate the parameters of computationally expensive models. However, the STD-BC algorithm does not scale well with regard to the number of data points

In this paper, we introduce a new locally implicit sonic point processing algorithm for one-dimensional shallow-water equations. The algorithm is based on transferring the shallow-water invariants along the characteristics around the sonic point, and it can be coupled with conservative-characteristic methods, which have problems modelling the transonic flows. For each sonic point, a system of two non-linear

The data-centric construction of inexpensive surrogates for fine-grained, physical models has been at the forefront of computational physics due to its significant utility in many-query tasks such as uncertainty quantification. Recent efforts have taken advantage of the enabling technologies from the field of machine learning (e.g., deep neural networks) in combination with simulation data. While such

An arbitrary order finite difference method for curved boundary domains with Cartesian grid is proposed. The technique handles in a universal manner Dirichlet, Neumann or Robin conditions. We introduce the Reconstruction Off-site Data (ROD) method, that transfers in polynomial functions the information located on the physical boundary to the computational domain. Three major advantages are: (1) a simple

An asynchronous task-based Eulerian-Lagrangian approach for efficient parallel multi-physics simulations that can scale for arbitrary large number of particles and non-uniformly distributed particles is presented. The parallel methodology is based on a task-based partitioning of the multi-physics problem, where each single-physics problem is considered as a task and carried out using its own set of

In this paper, we study a parallel-in-time (PinT) algorithm for all-at-once system from a non-local evolutionary equation with weakly singular kernel where the temporal term involves a non-local convolution with a weakly singular kernel and the spatial term is the usual Laplacian operator with variable coefficients. Such a problem has been intensively studied in recent years thanks to the numerous

This article aims at developing a high order pressure-based solver for the solution of the 3D compressible Navier-Stokes system at all Mach numbers. We propose a cell-centered discretization of the governing equations that splits the fluxes into a fast and a slow scale part, that are treated implicitly and explicitly, respectively. A novel semi-implicit discretization is proposed for the kinetic energy

In this paper, we present recent improvements of an Immersed Boundary Method (IBM) for the simulation of turbulent compressible flows on Cartesian grids. The proposed approach enables to remove spurious oscillations at the wall on skin pressure and friction coefficients. Results are compared to a body-fitted approach using the same wall function, showing that the stair-step immersed boundary provides

Particle-in-cell codes are the most widely used simulation tools for kinetic studies of ultra-intense laser-plasma interactions. Using the motion of a single electron in a plane electromagnetic wave as a benchmark problem, we show surprising deterioration of the numerical accuracy of the PIC algorithm with increasing normalized wave amplitude for typical time-step and grid sizes. Two significant sources

Deep learning is a powerful tool for solving nonlinear differential equations, but usually, only the solution corresponding to the flattest local minimizer can be found due to the implicit regularization of stochastic gradient descent. This paper proposes a network-based structure probing deflation method to make deep learning capable of identifying multiple solutions that are ubiquitous and important

We propose an Hermite spectral method for the Fokker-Planck-Landau (FPL) equation. Both the distribution functions and the collision terms are approximated by series expansions of the Hermite functions. To handle the complexity of the quadratic FPL collision operator, a reduced collision model is built by adopting the quadratic collision operator for the lower-order terms and the diffusive Fokker-Planck

An implicit and conservative numerical scheme is proposed for the isotropic quantum Fokker-Planck equation describing the evolution of degenerate electrons subject to elastic collisions with other electrons and ions. The electron-ion and electron-electron collision operators are discretized using a discontinuous Galerkin method, and the electron energy distribution is updated by an implicit time integration

In this paper, an energy-consistent finite difference scheme for the compressible hydrodynamic and magnetohydrodynamic (MHD) equations is introduced. For the compressible magnetohydrodynamics, an energy-consistent finite difference formulation is derived using the product rule for the spatial difference. The conservation properties of the internal, kinetic, and magnetic energy equations can be satisfied

We make a comparison of the finite difference and multiresolution method for solving the elliptic equations on irregular domains. The Dirichlet boundary condition is treated by the ghost fluid method (GFM) for the finite difference method and the Lagrange multiplier for the multiresolution method. Numerical results illustrate the improved convergence rate of errors and their gradients with the multiresolution

In this paper we present a mathematical model and a numerical workflow for the simulation of a thermal single-phase flow with reactive transport in porous media, in the presence of fractures. The latter are thin regions which might behave as high or low permeability channels depending on their physical parameters, and are thus of paramount importance in underground flow problems. Chemical reactions

This work presents, for the first time, a dual time stepping (DTS) approach to solve the global system of equations that appears in the hybridisable discontinuous Galerkin (HDG) formulation of convection-diffusion problems. A proof of the existence and uniqueness of the steady state solution of the HDG global problem with DTS is presented. The stability limit of the DTS approach is derived using a

This study presents an approach to discover the significant terms in partial differential equations (PDEs) that describe particular phenomena based on the measured data. The relationship between the known field data and its continuous representation of PDEs is achieved through a linear regression model. It specifically employs the peridynamic differential operator (PDDO) and sparse linear regression

Statistical (machine learning) tools for equation discovery require large amounts of data that are typically computer generated rather than experimentally observed. Multiscale modeling and stochastic simulations are two areas where learning on simulated data can lead to such discovery. In both, the data are generated with a reliable but impractical model, e.g., molecular dynamics simulations, while

Simulations of turbulent fluid flow around long cylindrical structures are computationally expensive because of the vast range of length scales, requiring simplifications such as dimensional reduction. Current dimensionality reduction techniques such as strip-theory and depth-averaged methods do not take into account the natural flow dissipation mechanism inherent in the small-scale three-dimensional

Detailed description of the transport processes in plasma is crucial for many disciplines. When the mean-free-path of the electrons is comparable or exceeds a characteristic length scale of the plasma profile, non-local behavior can be observed. Predictions of the diffusion theory are not valid and non-local electric and magnetic fields are generated. Kinetic modeling of these phenomena on time scales

An analysis of calibration for reduced-order models (ROMs) is presented in this work. The Galerkin and least-squares Petrov-Galerkin (LSPG) methods are tested on compressible flows involving a disparity of temporal scales. A novel calibration strategy is proposed for the LSPG method and two test cases are analyzed. The first consists of a subsonic airfoil flow where boundary layer instabilities are

In this work we address the compressible Navier-Stokes equations written in the so-called primitive formulation. The proposed methodology is a finite-element solver based on a fractional step scheme in time, which allows to uncouple the calculation of the problem unknowns providing important savings in computational cost. In addition, we include a stabilization technique within the Variational Multi-Scale

Computational challenges arise for the immersed boundary method (IBM) when dealing with compressible flow, where discontinuous and smoothly varying flow regions appear near the immersed boundary. The conventional ghost-cell IBM provides inaccurate results for smoothly varying regions when a low-order interpolation is used, or it suffers from a numerical instability for the discontinuous flow when a

We introduce the concept of a Graph-Informed Neural Network (GINN), a hybrid approach combining deep learning with probabilistic graphical models (PGMs) that acts as a surrogate for physics-based representations of multiscale and multiphysics systems. GINNs address the twin challenges of removing intrinsic computational bottlenecks in physics-based models and generating large data sets for estimating

We introduce a new variational numerical method which does not require any background mesh to compute the scheme coefficients. Replacing the mesh by a point cloud endowed with connectivity, which we call a discretization network, and following the virtual element framework we derive a consistent, coercive and stable numerical scheme. We illustrate the good behavior of the method on several numerical

A grouping-circular-based (GCB) greedy algorithm is proposed to improve the efficiency of mesh deformation. By incorporating the multigrid concept that the computational errors on the fine mesh can be approximated with those on the coarse mesh, this algorithm stochastically divides all boundary nodes into m groups and uses the locally maximum radial basis functions (RBF) interpolation error of the

In this work, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. While the equations exhibit diversified solutions from various

This paper constructs adaptive sparse grid collocation method onto arbitrary order piecewise polynomial space. The sparse grid method is a popular technique for high dimensional problems, and the associated collocation method has been well studied in the literature. The contribution of this work is the introduction of a systematic framework for collocation onto high-order piecewise polynomial space

In this article, we apply direct discontinuous Galerkin method with interface correction (DDGIC) to solve chemotaxis Keller-Segel equation and prove the quadratic polynomial solution satisfying positivity-preserving with third order of accuracy. We show DDGIC method can obtain optimal convergence order for the cell density variable without introducing extra auxiliary variables to approximate the gradient

A height function based numerical scheme has been developed to apply contact angle in the parallel computational framework of 3D volume-of-fluid method. Given that most of the reported height function methods are implemented in 2D, the differences between the implementations of 2D and 3D height function methods are first focused. Then, a novel scheme is proposed to preserve the accuracy and consistency

MILU preconditioner is well known [16], [3] to be the optimal choice among all the ILU-type preconditioners in solving the Poisson equation with Dirichlet boundary conditions. However, it is less known which is an optimal preconditioner in solving the Poisson equation with Neumann boundary conditions. The condition number of an unpreconditioned matrix is as large as O(h−2), where h is the step size

Inverse modeling for computing a high-dimensional spatially-varying property field from indirect sparse and noisy observations is a challenging problem. This is due to the complex physical system of interest often expressed in the form of multiscale PDEs, the high-dimensionality of the spatial property of interest, and the incomplete and noisy nature of observations. To address these challenges, we

We present a STRUcture-Preserving HYbrid code - STRUPHY - for the simulation of magneto-hydrodynamic (MHD) waves interacting with a population of energetic particles far from thermal equilibrium (kinetic species). The implemented model features linear, ideal MHD equations in curved, three-dimensional space, coupled nonlinearly to the full-orbit Vlasov equations via a current coupling scheme. The algorithm

A novel implementation route of the wall-function method to the lattice Boltzmann method (LBM) is proposed to extend the applicability of the LBM for high Reynolds number turbulent heat transfer in complex geometries. The proposed immersed virtual wall method assumes the virtual wall layer beneath the wall which satisfies the slip wall conditions allowing the subsurface heat and fluid flows within

An accurate and efficient normalized gradient flow (NGF) method is developed to compute the ground state of a spinor Bose-Einstein condensate (BEC) with an arbitrary positive integer spin, which is described as a vector wave function that minimizes the Gross-Pitaevskii energy functional under the two constraints of the total mass and magnetization. Similar to the NGF for computing the ground states

We propose a moment limiter of arbitrary high order for the discontinuous Galerkin method on unstructured triangular meshes. The limiter works by hierarchically limiting solution coefficients (moments) by comparing them to reconstructed directional derivatives along specific directions. Limiting along these directions is performed using one-dimensional minmod slope limiters. The stencil used to reconstruct

We introduce a new numerical method for the time-dependent Maxwell equations on unstructured meshes in two space dimensions. This relies on the introduction of a new mesh, which is the barycentric-dual cellular complex of the starting simplicial mesh, and on approximating two unknown fields with integral quantities on geometric entities of the two dual complexes. A careful choice of basis functions

This manuscript presents a new extended linear system for integral equation based techniques for solving boundary value problems on locally perturbed geometries. The new extended linear system is similar to a previously presented technique for which the authors have constructed a fast direct solver. The key features of the work presented in this paper are that the fast direct solver is more efficient

A friction model for hyper-elastic solid materials was proposed for the simulation of complex shear impacts in a Eulerian framework. The interfacial status of multi-material interactions was obtained in Harten, Lax, and van Leer discontinuity (HLLD) Riemann solver. The inverse deformation gradient tensor was utilized in governing equations to describe the shape change of hyper-elastic solid materials

We present a finite element based variational interface-preserving and conservative phase-field formulation for the modeling of incompressible two-phase flows with surface tension dynamics. The preservation of the hyperbolic tangent interface profile of the convective Allen-Cahn phase-field formulation relies on a novel time-dependent mobility model. The mobility coefficient is adjusted adaptively

We present a novel approach aimed at high-performance uncertainty quantification for time-dependent problems governed by partial differential equations. In particular, we consider input uncertainties described by a Karhunen-Loève expansion and compute statistics of high-dimensional quantities-of-interest, such as the cardiac activation potential. Our methodology relies on a close integration of multilevel

We introduce a domain decomposition structure-preserving method based on a hybrid mimetic spectral element method for three-dimensional linear elasticity problems in curvilinear conforming structured meshes. The method is an equilibrium method which satisfies pointwise equilibrium of forces. The domain decomposition is established through hybridization which first allows for an inter-element normal

In this paper, a multiscale and monolithic arbitrary Lagrangian–Eulerian finite element method (ALE-FEM) is developed for a multiscale hemodynamic fluid-structure interaction (FSI) problem involving an aortic aneurysm growth to quantitatively predict the long-term aneurysm risk in the cardiovascular environment, where the blood fluid profile, the hyperelastic arterial wall, and the aneurysm pathophysiology

We present an adaptive moving mesh method for unstructured meshes which is a three-dimensional extension of the previous works of Ceniceros et al. [10], Tang et al. [40] and Chen et al. [11]. The iterative solution of a variable diffusion Laplacian model on the reference domain is used to adapt the mesh to moving sharp solution fronts while imposing slip conditions for the displacements on curved boundary

Perfectly Matched Layers (PMLs) appear as a popular alternative to non-reflecting boundary conditions for wave-type problems. The core idea is to extend the computational domain by a fictitious layer with specific absorption properties such that the wave amplitude decays significantly and does not produce back reflections. In the context of convected acoustics, it is well-known that PMLs are exposed

Production of hydrocarbons and water from subsurface reservoirs are known to cause permanent deformation of the reservoir and seismicity along faults both of which are detrimental to sustainable development of natural resources. Most of the prior studies on understanding fluid flow-induced plasticity and seismicity have focused on one or the other phenomenon due to the numerical difficulty associated

Multi-fidelity optimization methods promise a high-fidelity optimum at a cost only slightly greater than a low-fidelity optimization. This promise is seldom achieved in practice, due to the requirement that low- and high-fidelity models correlate well. In this article, we propose an efficient bi-fidelity shape optimization method for turbulent fluid-flow applications with Large-Eddy Simulation (LES)

Mixing phenomena are important mechanisms controlling flow, species transport, and reaction processes in fluids and porous media. Accurate predictions of reactive mixing are critical for many Earth and environmental science problems such as contaminant fate and remediation, macroalgae growth, and plankton biomass growth. To investigate the evolution of mixing dynamics under different scenarios (e.g

The Lagrangian-Averaged Navier-Stokes-α (LANS-α) turbulence model was implemented for the first time in a coastal hydrodynamic model. We present in this paper the details of the implementation, as well as the difficulties encountered. To overcome the difficulties, a convolution filter was used instead of the Helmoltz operator, and incompressibility was imposed in both rough and smooth velocities. The

The dissolution of solids has created spectacular geomorphologies ranging from centimeter-scale cave scallops to the kilometer-scale “stone forests” of China and Madagascar. Mathematically, dissolution processes are modeled by a Stefan problem, which describes how the motion of a phase-separating interface depends on local concentration gradients, coupled to a fluid flow. Simulating these problems

Bayesian approaches have been successfully integrated into training deep neural networks. One popular family is stochastic gradient Markov chain Monte Carlo methods (SG-MCMC), which have gained increasing interest due to their ability to handle large datasets and the potential to avoid overfitting. Although standard SG-MCMC methods have shown great performance in a variety of problems, they may be

This paper analyzes three formulations of the tensor artificial viscosity in two-dimensional Cartesian geometry, namely, Campbell-Shashkov viscosity, Lipnikov-Shashkov viscosity and Wendroff viscosity. We first present a general derivation of these three ones. The discrepancy between Campbell-Shashkov viscosity and Lipnikov-Shashkov viscosity is then provided, followed by the non-triviality proof of

All discretized numerical models contain modeling errors – this reality is amplified when reduced-order models are used. The ability to accurately approximate modeling errors informs statistics on model confidence and improves quantitative results from frameworks using numerical models in prediction, tomography, and signal processing. Further to this, the compensation of highly nonlinear and non-Gaussian

Discrete mechanics is presented as an alternative to the equations of fluid mechanics, in particular to the Navier-Stokes equation. The derivation of the discrete equation of motion is built from the intuitions of Galileo, the principles of Galilean equivalence and relativity. Other more recent concepts such as the equivalence between mass and energy and the Helmholtz-Hodge decomposition complete the

Based on the Allen-Cahn equation, we propose a phase-field model for liquid-vapor phase-change phenomena. We first extend the conservative form of the Allen-Cahn equation to include phase-change effects, and then develop a lattice-Boltzmann model to numerically solve the governing equations for interface motion and hydrodynamics, both of which include mass transfer effects. The net heat flux at the