This paper is devoted to an emerging research topic, which is the numerical computation of stationary states of a generic Dirac operator with nonlinear potential. We are more specifically interested in the numerical computation of chemical potentials and eigenenergies. In this goal, several approaches are explored namely Feit-Fleck's, Rayleigh-Ritz, and min-max method for the computation of chemical

The efficient numerical solution of many kinetic models in plasma physics is impeded by the stiffness of these systems. Exponential integrators are attractive in this context as they remove the CFL condition induced by the linear part of the system, which in practice is often the most stringent stability constraint. In the literature, these schemes have been found to perform well, e.g., for drift-kinetic

We present here a multilayer model for shallow grain-fluid mixtures with dilatancy effects. It can be seen as a generalization of the depth-averaged model presented in [Bouchut et al. A two-phase two-layer model for fluidized granular flows with dilatancy effects. J. Fluid Mech., 801:166-221, 2016], that includes dilatancy effects by considering a two-layer model, a mixture grain-fluid layer and an

Geometrical Volume-of-Fluid (VoF) methods mainly support structured meshes, and only a small number of contributions in the scientific literature report results with unstructured meshes and three spatial dimensions. Unstructured meshes are traditionally used for handling geometrically complex solution domains that are prevalent when simulating problems of industrial relevance. However, three-dimensional

We present a new flux-fixup approach for arbitrarily high-order discontinuous Galerkin (DG) discretizations of the SN transport equation, and we demonstrate the compatibility of this approach with the Variable Eddington Factor (VEF) method [1], [2]. The new fixup approach is sweep-compatible: during a transport sweep (block Gauss-Seidel iteration in which the scattering source is lagged), a local quadratic

We propose and analyse a fully adaptive strategy for solving elliptic PDEs with random data in this work. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost. The novel aspect of our strategy is that the hierarchy of spatial approximations

In this paper, we generalize the exponential energy-preserving integrator proposed in the recent paper [SIAM J. Sci. Comput. 38(2016) A1876-A1895] for conservative systems, which now becomes linearly implicit by further utilizing the idea of the scalar auxiliary variable approach. Comparing with the original exponential energy-preserving integrator which usually leads to a nonlinear algebraic system

The paper describes a flexible computational method that can be used to represent the complex geometric evolution of a fluid-driven fracture front using an unstructured triangular element mesh. A specific motivation for the method is to allow subsequent treatment of non-planar fracture growth problems. Elastic interactions between the crack opening displacements and the surrounding medium are calculated

A new computational strategy is proposed for determining all elastic scatterer characteristics including the shape, the material properties (Lamé coefficients and density), and the location from the knowledge of far-field pattern (FFP) measurements. The proposed numerical approach is a multi-stage procedure in which a carefully designed regularized iterative method plays a central role. The adopted

We review and extend the list of stability and convergence properties satisfied by Runge-Kutta (RK) methods that are associated with Summation-By-Parts (SBP) operators, herein called RK-SBP methods. The analysis covers classical, generalized as well as upwind SBP operators. Previous work on the topic has relied predominantly on energy estimates. In contrast, we derive all results using a purely algebraic

In this work, we propose a unified gas kinetic particle (UGKP) method for solving the nonlinear thermal radiative transfer equations. The UGKP method is a multiscale method in that the macro and microscopic variables are coupled and updated in a consistent way. It employs a finite volume formulation for the macroscopic variable evolution, and a particle-based Monte Carlo solver for tracking the non-equilibrium

In the literature on nonlinear, projection-based model order reduction for computational fluid dynamics problems, it is often claimed that due to modal truncation, a projection-based reduced-order model (PROM) does not resolve the dissipative regime of the turbulent energy cascade and therefore is numerically unstable. Efforts at addressing this claim have ranged from attempting to model the effects

In this work we present a new high order space-time discretization method based on a discontinuos Galerkin paradigm for the second order visco-elastodynamics equation. After introducing the method, we show that the resulting space-time discontinuous Galerkin formulation is well-posed, stable and retains optimal rate of convergence with respect to the discretization parameters, namely the mesh size

We consider the solution of the fully kinetic (including electrons) Vlasov-Ampère system in a one-dimensional physical space and two-dimensional velocity space (1D-2V) for an arbitrary number of species with a time-implicit Eulerian algorithm. The problem of velocity-space meshing for disparate thermal and bulk velocities is dealt with by an adaptive coordinate transformation of the Vlasov equation

Data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a data-based, probabilistic perspective that enables the quantification of predictive uncertainties. One of the outstanding problems has been the introduction of physical

The Poisson-Nernst-Planck (PNP) model is frequently-used in simulating ion transport through ion channel systems. Due to the nonlinear nature of the coupled system, choosing appropriate initial values is often crucial to helping obtain a convergent result or accelerate the convergence rate of finite element solution, especially for large channel protein systems. Continuation is an effective and commonly

In this paper, we present and study discontinuous Galerkin (DG) methods for one-dimensional multi-symplectic Hamiltonian partial differential equations. We particularly focus on semi-discrete schemes with spatial discretization only, and show that the proposed DG methods can simultaneously preserve the multi-symplectic structure and energy conservation with a general class of numerical fluxes, which

This paper is concerned with the boundary treatment of the lattice Boltzmann method (LBM) with general propagation for the incompressible Navier-Stokes equations. Since the propagation step involves three grid points, constructing effective boundary schemes for such method is highly nontrivial. To address this problem, we first construct an auxiliary boundary scheme under the diffusive scaling by using

Extension of the self-similar structures based genuinely two-dimension Riemann solver called MuSIC (Multidimensional, Self-similar, strongly-Interacting, Consistent) to curvilinear coordinate systems are conducted in this study. Built upon Balsara's work, the MuSIC1 (MuSIC considering the 1st order moment) scheme solves the compressible Euler equations in curvilinear coordinates by considering the

A filtering operation, based on B-spline discretizations, is introduced to target weakly growing mesh-scale oscillations that can arise in high-fidelity turbulence simulations. This is a spectral regularization that can be described using the singular values of a banded matrix operator, with the filtering strength set by a scalar- or vector-valued penalty parameter. The penalty parameter can be specified

The issue of the relaxation to equilibrium has been at the core of the kinetic theory of rarefied gas dynamics. In the paper, we introduce the Deep Neural Network (DNN) approximated solutions to the kinetic Fokker-Planck equation in a bounded interval and study the large-time asymptotic behavior of the solutions and other physically relevant macroscopic quantities. We impose the varied types of boundary

Smoothness indicator is an essential part of weighted essentially non-oscillatory (WENO) scheme, whose target is to distinguish discontinuous profiles and continuous profiles. However, the magnitudes of most of the smoothness indicators will be decreased significantly when the stencil approaches critical points, which would be negative for the numerical simulation. The reason should be that the first

We present a paradigm for developing arbitrarily high order, linear, unconditionally energy stable numerical algorithms for general gradient flow models. We apply the energy quadratization (EQ) technique to reformulate the gradient flow model into an equivalent one with a quadratic free energy and a modified mobility. For any positive integer k>0 and tn=nΔt, where Δt is the time step size, we linearize

We present a novel numerical scheme for the efficient and accurate solution of the isothermal two-fluid (electron + ion) equations coupled to Poisson's equation for low-temperature plasmas at low-pressure. The model considers electrons and ions as separate fluids, comprising the electron inertia and space charge regions. The discretization of this system with standard explicit schemes is constrained

We develop a new Lagrangian approach — flow dynamic approach to effectively capture the interface in the Allen-Cahn type equations. The underlying principle of this approach is the Energetic Variational Approach (EnVarA), motivated by Rayleigh and Onsager [27], [28]. Its main advantage, comparing with numerical methods in Eulerian coordinates, is that thin interfaces can be effectively captured with

We report on simulations of two-phase flows with deforming interfaces at various density contrasts by solving thermodynamically consistent Cahn-Hilliard Navier-Stokes equations. An (essentially) unconditionally energy-stable Crank-Nicolson-type time integration scheme is used. Detailed proofs of energy stability of the semi-discrete scheme and for the existence of solutions of the advective-diffusive

In this paper, we introduce a new extended version of the shallow-water equations with surface tension which may be decomposed into a hyperbolic part and a second order derivative part which is skew-symmetric with respect to the L2 scalar product. This reformulation allows for large gradients of fluid height simulations using a splitting method. This result is a generalization of the results published

The deformations of flagella are important in the motility of single- and multi-flagellated bacteria. Existing numerical methods have treated flagella as extensible filaments with a large extensional modulus, resulting in a stiff numerical problem and long simulation times. However, flagella are nearly inextensible, so to avoid large extensional stiffness, we introduce inextensible elastic rod models

In this paper, a new Cartesian grid finite difference method is introduced based on the fourth order accurate matched interface and boundary (MIB) method and fast Fourier transform (FFT). The proposed augmented MIB method consists of two major parts for solving elliptic interface problems in two dimensions. For the interior part, in treating a smoothly curved interface, zeroth and first order jump

We propose an algorithm for general nonlinear eigenvalue problems to compute physically relevant eigenvalues within a chosen contour. Eigenvalue information is explored by contour integration incorporating different weight functions. The gathered information is processed by solving a nonlinear system of equations of small dimension prioritizing eigenvalues with high physical impact. No auxiliary functions

We construct a divergence-free HDG scheme for the Cahn-Hilliard-Navier-Stokes phase-field model. The scheme is robust in the convection-dominated regime, produces a globally divergence-free velocity approximation, and can be efficiently implemented via static condensation. Two numerical benchmark problems, namely the rising bubble problem, and the Rayleigh-Taylor instability problem are used to show

This paper proposes a deep-learning-initialized iterative method (Int-Deep) for low-dimensional nonlinear partial differential equations (PDEs). The corresponding framework consists of two phases. In the first phase, an expectation minimization problem formulated from a given nonlinear PDE is approximately resolved with mesh-free deep neural networks to parametrize the solution space. In the second

A fundamental problem in geostatistical modeling is to infer the heterogeneous geological field based on limited measurements and some prior spatial statistics. Semantic inpainting, a technique for image processing using deep generative models, has been recently applied for this purpose, demonstrating its effectiveness in dealing with complex spatial patterns. However, the original semantic inpainting

We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic representation of the PDE in the spirit of the Feynman-Kac formula. The solution is given by an expectation of a martingale process driven by a Brownian motion. As Brownian

Learning nonlinear turbulent dynamics from partial observations is an important and challenging topic. In this article, an efficient learning algorithm based on the expectation-maximization approach is developed for a rich class of complex nonlinear turbulent dynamics using short training data. Despite the significant nonlinear and non-Gaussian features in these models, the analytically solvable conditional

Accurate representation and estimation of seismic sources is crucial to the joint medium-source full waveform inversion problem. We focus on the source estimation subproblem and its difficulties, where seismic sources are modeled by truncated series of multipoles. The source full waveform inversion formulation results in a highly ill-conditioned (potentially ill-posed) linear least squares problem

In this paper, we develop an effective numerical algorithm that tracks the trajectory needed to solve guiding center models using the backward semi-Lagrangian method. In terms of numerical calculations, two appreciably fast algorithms for the departure points are designed. One is a completely explicit formula for numerical solutions of the discrete system for each Cauchy problem. This formula is characterized

The paper addresses the numerical implementation of the multipole expansion method in application to the conductivity problem for the ellipsoidal particle composite with isotropic constituents and imperfect interfaces. The main steps of the numerical algorithm are considered in detail including generation of the geometry model, evaluating the re-expansion coefficients, fulfilling the periodicity and

In this paper we extensively study the stochastic Galerkin scheme for uncertain systems of conservation laws, which appears to produce oscillations already for a simple example of the linear advection equation with Riemann initial data. Therefore, we introduce a modified scheme that we call the weighted essentially non-oscillatory (WENO) stochastic Galerkin scheme, which is constructed to prevent the

Maximum-Entropy Distributions offer an attractive family of probability densities suitable for moment closure problems. Yet finding the Lagrange multipliers which parametrize these distributions, turns out to be a computational bottleneck for practical closure settings. Motivated by recent success of Gaussian processes, we investigate the suitability of Gaussian priors to approximate the Lagrange multipliers

The present study focuses on the unphysical effects induced by the use of non-uniform grids in the lattice Boltzmann method. In particular, the convection of vortical structures across a grid refinement interface is likely to generate spurious noise that may impact the whole computation domain. This issue becomes critical in the case of aeroacoustic simulations, where accurate pressure estimations

Due to the butterfly-effect, computer-generated chaotic simulations often deviate exponentially from the true solution, so that it is very hard to obtain a reliable simulation of chaos in a long-duration time. In this paper, a new strategy of the so-called Clean Numerical Simulation (CNS) in physical space is proposed for spatio-temporal chaos, which is computationally much more efficient than its

A new approach for the increase in the order of accuracy of high order elements used for the time dependent heat equation and for the time independent Poisson equation has been suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding discrete stencil equation with respect to the local truncation error. By a simple modification of the

This paper presents a quantitative dispersion-dissipation and stability analysis of the triangle-based discontinuous Galerkin method (DGM) for simulating elastic wave propagation. The analysis is carried out for both P- and S-waves, with semi-discrete and fully discrete cases. The DGM is based on the 1st-order hyperbolic system with numerical flux formulations. The semi-discrete analysis is considered

In this paper, we introduce some new high-order discrete formulations on general unstructured meshes, especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations. Working with a recent family of optimized asymptotically equivalent equations, we benefit from the simplified

This paper concentrates on numerical shock instabilities of some approximate Riemann solvers which stem from the HLLE method, including HLLC-type and HLLEM-type flux functions. Carbuncle phenomena are illustrated for these numerical schemes, especially for those believed to be carbuncle-free, such as the HLLE scheme. A linear matrix stability analysis shows that there exists a threshold independent

An augmented immersed finite element method is proposed for solving elliptic interface problems with discontinuous and variable coefficients. By introducing the jump of normal derivative of the solution along the interface as an augmented variable, the interface problem is solved iteratively on Cartesian meshes using the generalized minimal residual (GMRES) iterative method. Within each iteration,

We develop a level-set method in the finite-element framework. The contact line singularity is removed by the slip boundary condition proposed by Ren and E (2007) [6], which has two friction coefficients: βN that controls the slip between the bulk fluids and the solid wall and βCL that controls the deviation of the microscopic dynamic contact angle from the static one. The predicted contact line dynamics

Polynomial Chaos (PC) is one of the popular methods for approximate evaluation of responses of stochastic structural mechanics problems. In the case of a structural system under dynamic loading, the PC method is combined with a time integration scheme. The PC method with Galerkin projection converts the stochastic PDE to a set of simultaneous deterministic PDE, which can be solved using any standard

We present a new implicit asymptotic preserving time integration scheme for charged-particle orbit computation in arbitrary electromagnetic fields. The scheme is built on the Crank-Nicolson integrator and continues to recover full-orbit motion in the small time-step limit, but also recovers all the first-order guiding center drifts as well as the correct gyroradius when stepping over the gyration time-scale

A unified framework of invariant-conserving explicit Runge-Kutta schemes for the nonlinear Hamiltonian ODEs and PDEs are proposed by utilizing the invariant energy quadratization technique. First, the nonlinear Hamiltonian differential equation is transformed into an equivalent reformulation which admits a quadratic energy invariant. For the nonlinear ODE, the reformulation is then discretized using

This paper presents a high-order discontinuous Galerkin (DG) scheme for the simulation of wave propagation through coupled elastic-acoustic media. We use a first-order stress-velocity formulation, and derive a simple upwind-like numerical flux which weakly imposes continuity of the normal velocity and traction at elastic-acoustic interfaces. When combined with easily invertible weight-adjusted mass

The complexity of molecular dynamics simulations necessitates dimension reduction and coarse-graining techniques to enable tractable computation. The generalized Langevin equation (GLE) describes coarse-grained dynamics in reduced dimensions. In spite of playing a crucial role in non-equilibrium dynamics, the memory kernel of the GLE is often ignored because it is difficult to characterize and expensive

We present a novel methodology for accurate treatment of viscous terms in evaporation problems. The proposed scheme is an extension of the sharp viscous treatment of Kang et al. (2000) [7] to 3D phase change problems. To ensure accuracy and grid converging solutions, a new implicit approach to computing viscous fluxes across the phase interfaces is proposed, which was previously unavailable in fixed

We present an energy-stable scheme for simulating the incompressible Navier-Stokes equations based on the generalized Positive Auxiliary Variable (gPAV) framework. In the gPAV-reformulated system the original nonlinear term is replaced by a linear term plus a correction term, where the correction term is put under control by an auxiliary variable. The proposed scheme incorporates a pressure-correction

This article presents a semi-discrete, multilayer set of equations describing the three-dimensional motion of an incompressible fluid bounded below by topography and above by a moving free-surface. This system is a consistent discretisation of the incompressible Euler equations, valid without assumptions on the slopes of the interfaces. Expressed as a set of conservation laws for each layer, the formulation

The discontinuous Galerkin (DG) methods have attained increasing popularity for solving the incompressible Navier-Stokes (INS) equations in recent years. However, the DG methods have their own weakness due to the high computational costs and storage requirements. In this work, we develop a high-order hybrid reconstructed DG (rDG) method for solving the INS equations on arbitrary grids. To be specific

In this paper we present a numerical approach to solve the Navier-Stokes equations on moving domains with second-order accuracy. The space discretization is based on the ghost-point method, which falls under the category of unfitted boundary methods, since the mesh does not adapt to the moving boundary. The equations are advanced in time by using Crank-Nicholson. The momentum and continuity equations

A semi-implicit finite difference time domain (FDTD) numerical Maxwell solver is developed for full electromagnetic Particle-in-Cell (PIC) codes for the simulations of plasma-based acceleration. The solver projects the volumetric Yee lattice into planes transverse to a selected axis (the particle acceleration direction). The scheme - by design - removes the numerical dispersion of electromagnetic waves