In this article, we propose high order discontinuous Galerkin entropy stable schemes for ten-moment Gaussian closure equations, based on the suitable quadrature rules (see [1]). The key components of the proposed schemes are the use of an entropy conservative numerical flux [2] in each cell and an appropriate entropy stable numerical flux at the cell edges. These fluxes are then used in the entropy

In this paper, a new numerical computational frame is presented for solving moving interface problems modeled by parabolic PDEs on static and evolving surfaces. The surface PDEs can have Dirac delta source distributions and discontinuous coefficients. One application is for thermally driven moving interfaces on surfaces such as Stefan problems and dendritic solidification phenomena on solid surfaces

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 evolution. To investigate the evolution of mixing dynamics under different scenarios (e

While a nonlinear viscosity is used widely to control oscillations when solving conservation laws using high-order elements based methods, such techniques are less straightforward to apply in global spectral methods as a local estimate of solution regularity generally is required. In this work we demonstrate how to train and use a local artificial neural network to estimate the local solution regularity

This work is dedicated to the solution filtering technique for performing direct and large-eddy simulation. It is shown that this approach is equivalent to the use of spectral viscosity as a possible ersatz of subgrid-scale modelling. In the framework of finite-difference schemes, the filter operator can be designed to ensure time consistency while easily controlling the level and scale selectivity

We consider a class of Fuchsian equations that, for instance, describes the evolution of compressible fluid flows on a cosmological spacetime. Using the method of lines, we introduce a numerical algorithm for the singular initial value problem when data are imposed on the cosmological singularity and the evolution is performed from the singularity hypersurface. We approximate the singular Cauchy problem

A framework for the numerical solution of the Landau-Lifshitz-Gilbert equation is developed in this paper. The numerical framework is based on the finite element method on tetrahedral meshes for the spatial discretization and the implicit midpoint scheme for the temporal discretization. The computational complexity for calculating the demagnetization field is effectively reduced by using a PDE approach

This paper introduces a new family of hybrid estimators aimed at controlling the efficiency of Monte Carlo computations in particle transport problems. In this context, efficiency is usually measured by the figure of merit (FOM) given by the inverse product of the estimator variance Var[ξ] and the run time T: FOM:=(Var[ξ]T)−1. Previously, we developed a new family of transport-constrained unbiased

A formal filtering process is used to simplify fields consisting of smooth layers by concentrating them into singular sheets, filaments and points. The approach is based on weighted coordinate smoothing, where the coordinates of nearby material points possessing non-zero values of the field are smoothed, using a nonlinear diffusion equation, thus moving them together. Examples for two and three-dimensional

This paper develops the high-order accurate entropy stable (ES) finite difference schemes for the shallow water magnetohydrodynamic (SWMHD) equations. They are built on the numerical approximation of the modified SWMHD equations with the Janhunen source term. First, the second-order accurate well-balanced semi-discrete entropy conservative (EC) schemes are constructed, satisfying the entropy identity

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

In this work we propose a high-order discretization of the Baer-Nunziato two-phase flow model (Baer and Nunziato, Int. J. Multiphase Flow, 12 (1986), pp. 861-889) with closures for interface velocity and pressure adapted to the treatment of discontinuous solutions, and stiffened gas equations of states. We use the discontinuous Galerkin spectral element method (DGSEM), based on collocation of quadrature

Numerical simulations of forced turbulence in compressible fluids are challenging due to the multi-scale nature of the problem and conflicting requirements for numerical methods to accurately resolve the small scales and, at the same time, to handle shock waves and other discontinuities without generating spurious oscillations. Minimizing nonlinear instability and aliasing error while maintaining high

Data-driven discovery of differential equations has been an emerging research topic. We propose a novel algorithm subsampling-based threshold sparse Bayesian regression (SubTSBR) to tackle high noise and outliers. The subsampling technique is used for improving the accuracy of the Bayesian learning algorithm. It has two parameters: subsampling size and the number of subsamples. When the subsampling

A system of mass balance equations is set up for plasma-chemical simulations, we derive the diffusion velocities from Stefan-Maxwell theory. An algorithm to linearly transform the system of mass balance equations in the context of non-LTE plasmas is described. This transformation is derived from the reaction set, and eliminates part of the chemical source term, enforces invariants such as conservation

In this paper we study finite element method for three-dimensional incompressible resistive magnetohydrodynamic equations, in which the velocity, the current density, and the magnetic induction are divergence-free. It is desirable that the discrete solutions should also satisfy divergence-free conditions exactly especially for the momentum equations. Inspired by constrained transport method, we devise

Knowing how the solution to time-harmonic wave scattering problems depends on medium properties and boundary conditions is pivotal in wave-based inverse problems, e.g. for imaging. This paper is devoted to the exposition of a computationally efficient method, called the adjoint state method, that allows to quantify the influence of media properties, directly and through boundary conditions, in the

We present a method for the data-driven learning of physical phenomena whose evolution in time depends on history terms. It is well known that a Mori-Zwanzig-type projection produces a description of the physical phenomena that depends on history, and also incorporates noise. If the data stream is sampled from the projected Mori-Zwanzig manifold, the description of the phenomenon will always depend

The time fractional Klein–Kramers model (TFKKM) is obtained by incorporating the subdiffusive mechanisms into the Klein–Kramers formalism. The TFKKM can efficiently express subdiffusion while an external force field is present in the phase space. The model describes the escape of a particle over a barrier and has a significant role in examining a variety of systems including slow (subdiffusion) dynamics

Data-driven surrogate models are widely used for applications such as design optimization and uncertainty quantification, where repeated evaluations of an expensive simulator are required. For most partial differential equation (PDE) simulations, the outputs of interest are often spatial or spatial-temporal fields, leading to very high-dimensional outputs. Despite the success of existing data-driven

We introduce DeepMoD, a Deep learning based Model Discovery algorithm. DeepMoD discovers the partial differential equation underlying a spatio-temporal data set using sparse regression on a library of possible functions and their derivatives. A neural network is used as function approximator and its output is used to construct the function library, allowing to perform the sparse regression within the

This study on overset meshes for incompressible-flow simulations is motivated by accurate prediction of wind farm aerodynamics involving large motions and deformations of components with complex geometry. Using first-order hyperbolic and elliptic equation proxies for the incompressible Navier-Stokes (NS) equations, we investigate the influence of information exchange between overset meshes on numerical

In this work, we consider the problem of optimal design of an acoustic cloak under uncertainty and develop scalable approximation and optimization methods to solve this problem. The design variable is taken as an infinite-dimensional spatially-varying field that represents the material property, while an additive infinite-dimensional random field represents, e.g., the variability of the material property

An evolving pressure projection method for numerical computations of the weakly compressible Navier-Stokes equations is proposed. Fully explicit time integration is achieved using an independent pressure evolution equation. To damp the acoustic wave in a weakly compressible fluid flow, we iteratively compute the pressure evolution equation coupled with a projection step. Due to the simplicity and locality

Data-driven turbulence modelling approaches are gaining increasing interest from the CFD community. Such approaches generally aim to improve the modelled Reynolds stresses by leveraging data from high fidelity turbulence resolving simulations. However, the introduction of a machine learning (ML) model introduces a new source of uncertainty, the ML model itself. Quantification of this uncertainty is

A novel interface method is developed in this paper for two-dimensional smoothed particle hydrodynamics (SPH) modellings of multiphase flows. The present interface method aims to resolve two essential issues in the multiphase flow simulations: the interface detection and the implementation of surface tension force. Specifically, a novel and easy-to-implement algebraic indicator is proposed to detect

We are interested in the development of an algorithmic differentiation framework for computing approximations to tangent vectors to scalar and systems of hyperbolic partial differential equations. The main difficulty of such a numerical method is the presence of shock waves that are resolved by proposing a numerical discretization of the calculus introduced in Bressan and Marson [Rend. Sem. Mat. Univ

Recently, we proposed a method to estimate parameters of stochastic dynamics based on the linear response statistics. The method rests upon a nonlinear least-squares problem that takes into account the response properties that stem from the Fluctuation-Dissipation Theory. In this article, we address an important issue that arises in the presence of model error. In particular, when the equilibrium density

The scalar field and the non-equilibrium solutions of the linear advection-diffusion d2Q9 Lattice Boltzmann (LBM) two-relaxation-times (TRT) scheme are constructed analytically. The scheme copes with an infinite number of suitable, second-order accurate, equilibrium weights. Here, the simplest, translation-invariant geometry with an implicitly located, straight or diagonal, grid-aligned interface (boundary)

In this paper, a ghost structure (GS) finite difference method is proposed to simulate the fractional FitzHugh-Nagumo (FHN) monodomain model on a moving irregular computational domain. In the GS formulation the moving irregular domain is converted into a fixed regular domain (called ghost structure), and the transmembrane potential is described in the Eulerian coordinates, while the membrane dynamics

Diagonalization-based parallel-in-time (PinT) implementation of linear multi-step methods has been well investigated in recent years. The main idea lies in first formulating the difference equations for all the time steps into an all-at-once algebraic system and then solving this system iteratively with a block α-circulant preconditioner. It is noticed that the construction of the preconditioner is

Recently, the advent of deep learning has spurred interest in the development of physics-informed neural networks (PINN) for efficiently solving partial differential equations (PDEs), particularly in a parametric setting. Among all different classes of deep neural networks, the convolutional neural network (CNN) has attracted increasing attention in the scientific machine learning community, since

We present a uniformly first order accurate numerical method for solving the Klein-Gordon-Zakharov (KGZ) system with two dimensionless parameters 0<ε≤1 and 0<γ≤1, which are inversely proportional to the plasma frequency and the acoustic speed, respectively. In the simultaneous high-plasma-frequency and subsonic limit regime, i.e. ε<γ→0+, the KGZ system collapses to a cubic Schrödinger equation, and

This work introduces a novel method to represent the topological complexity of liners and boundaries in Low Order Models for thermoacoustic instabilities, under the assumption of zero-Mach number flow. In typical industrial combustion devices, the difficulty to model these elements is twofold: (1) they are characterized by complex-valued Rayleigh conductivities or acoustic impedances, and (2) they

We study semi-discrete central-upwind schemes and develop a new technique that allows one to decrease the amount of numerical dissipation present in these schemes without compromising their robustness. The goal is achieved by obtaining more accurate estimates for the one-sided local speeds of propagation using the discrete Rankine-Hugoniot conditions. In the two-dimensional case, these estimates are

In this work, we apply a fast and accurate numerical method for solving fractional reaction-diffusion equations in unbounded domains. By using the Fourier-like spectral approach in space, this method can effectively handle the fractional Laplace operator, leading to a fully diagonal representation of the fractional Laplacian. To fully discretize the underlying nonlinear reaction-diffusion systems,

Probabilistic inverse problems arise in a variety of scientific and engineering applications. A particular type of probabilistic inverse problem, termed a generalized aggregate data inverse problem, involves the specification of expected value targets and/or probabilistic constraints at discrete times or in a discrete set of transformed domains. Because the transformed distributions themselves are

In an ever-increasing interest for Machine Learning (ML) and a favorable data development context, we here propose an original methodology for data-based prediction of two-dimensional physical fields. Polynomial Chaos Expansion (PCE), widely used in the Uncertainty Quantification community (UQ), has long been employed as a robust representation for probabilistic input-to-output mapping. It has been

A moving-grid, shock-fitting, finite element method has been implemented that can achieve high-order accuracy for flow simulations with shocks. In this approach, element edges in the computational mesh are fitted to the shock front and moved with the shock throughout the simulation. The Euler or Navier-Stokes equations are solved on the moving mesh in an arbitrary Lagrangian-Eulerian framework. The

In this paper, we present an approach to deal with the dynamics of the Dirac equation with time-dependent electromagnetic potentials using the fourth-order compact time-splitting method (S4c). To this purpose, the time-ordering technique for time-dependent Hamiltonians is introduced, so that the influence of the time-dependence could be limited to certain steps which are easy to treat. Actually, in

In this work, we propose an efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solutions of two-dimensional (2-D) and three-dimensional (3-D) elliptic interface problems which may have discontinuous coefficients and curved interfaces with or without sharp corners. The proposed method uses Pascal polynomials as basis functions and utilizes multiple-scale

We present a discretization of the dynamics of sea-ice on triangular grids. Our numerical approach is based on the nonconforming Crouzeix-Raviart finite element. An advantage of this element is that it facilitates the coupling to an ocean model that employs an Arakawa C-type staggering of variables. We show that the Crouzeix-Raviart element implements a discretization of the viscous-plastic and el

Solving compressible flows containing both smooth and discontinuous flow structures still remains a big challenge for finite volume methods, especially on unstructured grids where one faces more difficulties in building high-order polynomial reconstruction and limiting projection to suppress numerical oscillations in comparison with the case of structured grids. As a result, most of the current finite

We present a new neural-network architecture, called the Cholesky-factored symmetric positive definite neural network (SPD-NN), for modeling constitutive relations in computational mechanics. Instead of directly predicting the stress of the material, the SPD-NN trains a neural network to predict the Cholesky factor of the tangent stiffness matrix, based on which the stress is calculated in incremental

We introduce an approach for solving the incompressible Navier-Stokes equations on a forest of Octree grids in a parallel environment. The methodology uses the p4est library of Burstedde et al. (2011) [15] for the construction and the handling of forests of Octree meshes on massively parallel distributed machines and the framework of Mirzadeh et al. (2016) [54] for the discretizations on Octree data

We present PFNN, a penalty-free neural network method, to efficiently solve a class of second-order boundary-value problems on complex geometries. To reduce the smoothness requirement, the original problem is reformulated to a weak form so that the evaluations of high-order derivatives are avoided. Two neural networks, rather than just one, are employed to construct the approximate solution, with one

In this article, we consider fast direct solvers for nonlocal operators. The pivotal idea is to combine a wavelet representation of the system matrix, yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The latter drastically reduces the fill-in during the factorization of the system matrix by means of a Cholesky decomposition or an LU decomposition, respectively. This way,

In this paper, a novel stabilized time-weighted residual methodology under the umbrella of Petrov-Galerkin time finite element formulation is developed to design a generalized computational framework, which permits unconditionally stable, high-order time accuracy, and features with controllable numerical dissipation for solving transient first-order systems. Various unconditionally stable (A/L-stable)

The Poisson-Nernst-Planck (PNP) equation is one important continuum model for studying charge transport in ion channels, which is a common phenomenon and plays a key role in molecular biosciences. In this paper, to improve the current PNP solvers, according to the equation structure, a new block preconditioner was proposed and proved that the preconditioned linear system has bounded eigenvalues independent

We present a real-space computational method called treecode-accelerated Green Iteration (TAGI) for all-electron Kohn-Sham Density Functional Theory. TAGI is based on a reformulation of the Kohn-Sham equations in which the eigenvalue problem in differential form is converted into a fixed-point problem in integral form by convolution with the modified Helmholtz Green's function. In each self-consistent

This work aims at identifying and quantifying uncertainties related to elastic and viscoelastic parameters, which characterize the arterial wall behavior, in one-dimensional modeling of the human arterial hemodynamics. The chosen uncertain parameters are modeled as random Gaussian-distributed variables, making stochastic the system of governing equations. The proposed methodology is initially validated

Deep Reinforcement Learning (DRL) has recently spread into a range of domains within physics and engineering, with multiple remarkable achievements. Still, much remains to be explored before the capabilities of these methods are well understood. In this paper, we present the first application of DRL to direct shape optimization. We show that, given adequate reward, an artificial neural network trained

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity

To minimise systematic errors in Monte Carlo simulations of charged particles, long range electrostatic interactions have to be calculated accurately and efficiently. Standard approaches, such as Ewald summation or the naive application of the classical Fast Multipole Method, result in a cost per Metropolis-Hastings step which grows in proportion to some positive power of the number of particles N

This paper presents a novel Interpolated Factored Green Function method (IFGF) for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF algorithm evaluates the action of Green function-based integral operators at a cost of O(Nlog⁡N) operations for an N-point surface mesh. The IFGF strategy, which leads

The objective of the present work is to develop a new numerical framework for simulations involving deformable domains, in the specific context of high-order meshes consistent with Computer-Aided Design (CAD) representations. Thus, the proposed approach combines ideas from isogeometric analysis, able to handle exactly CAD-based geometries, and Discontinuous Galerkin (DG) methods with an Arbitrary Lagrangian-Eulerian

In this paper, we extend our previous immersed boundary (IB) projection method for 2D inextensible vesicle in unsteady Stokes flow (Ong and Lai, 2020 [23]) to 3D incompressible interface as a prototype of vesicles. In spite of similar numerical algorithm to the 2D case, the present 3D numerical implementation is far from straightforward. An incompressible interface immersed in Newtonian fluid must

In this article we present a formulation for the numerical computation of static magnetic fields in discontinuous media. Focusing on solving magneto-fuid-structure interaction problems, we developed a finite volume multi-region framework capable of treating an arbitrary number of magnetized, permeable and current carrying bodies. The balance equations are written in terms of the magnetic vector potential

The paper is concerned with classic kernel interpolation methods, in addition to approximation methods that are augmented by gradient measurements. To apply kernel interpolation using radial basis functions (RBFs) in a stable way, we propose a type of quasi-optimal interpolation points, searching from a large set of candidate points, using a procedure similar to designing Fekete points or power function

The general synthetic iterative scheme (GSIS) is extended to find the steady-state solution of nonlinear gas kinetic equation, resolving the long-standing problems of slow convergence and requirement of ultra-fine grids in near-continuum flows. The key ingredient of GSIS is the tight coupling of gas kinetic and macroscopic synthetic equations, where the constitutive relations explicitly contain Newton's