Fixed-point iterative sweeping methods were developed in the literature to efficiently solve steady state solutions of Hamilton-Jacobi equations and hyperbolic conservation laws. Similar as other fast sweeping schemes, the key components of this class of methods are the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. A family of characteristics of the corresponding

We develop a boundary integral equation solver for elliptic partial differential equations on complex 3D geometries. Our method is efficient, high-order accurate and robustly handles complex geometries. A key component is our singular and near-singular layer potential evaluation scheme, hedgehog: a simple extrapolation of the solution along a line to the boundary. We present a series of geometry-processing

In this paper, we consider flow and transport problems in thin domains. Modeling problems in thin domains occur in many applications, where thin domains lead to some type of reduced models. A typical example is one dimensional reduced-order model for flows in pipe-like geometries (e.g., blood vessels). In many reduced-order models, the model equations are described apriori by some analytical approaches

We present a new disparity functional to measure and improve the geometric accuracy of a curved high-order mesh that approximates a target geometry model. To aim at real CAD models, We have devised the disparity to account for compound models, be independent of the entity parameterization, and allow trimmed entities. The disparity depends on the physical mesh and the auxiliary parametric meshes. Since

In this work, we propose a novel block preconditioner, labeled Explicit Decoupling Factor Approximation (EDFA), to accelerate the convergence of Krylov subspace solvers used to address the sequence of non-symmetric systems of linear equations originating from flow simulations in porous media. The flow model is discretized blending the Mixed hybrid finite element method for Darcy's equation with the

In solid-state physics, energies of crystals are usually computed with a plane-wave discretization of Kohn-Sham equations. However the presence of Coulomb singularities requires the use of large plane-wave cut-offs to produce accurate numerical results. In this paper, an analysis of the plane-wave convergence of the eigenvalues of periodic linear Hamiltonians with Coulomb potentials using the variational

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

In a recent work Feng, Liu and Wang (2020) [10], we imbedded characteristic compressing into an artificial neuron (AN) to propose a shock wave indicator on uniform mesh. In this work, the indicator is developed to unstructured grid. To achieve that, we retrain an AN on 1D randomly perturbed mesh, two prior information, (a) eigenvalue variable and (b) side-weighted average, is used in data pre-processing

This work outlines a new three-dimensional diffuse interface finite volume method for the simulation of multiple solid and fluid components featuring large deformations, sliding and void opening. This is achieved by extending an existing reduced-equation diffuse interface method by means of a number of novel flux-modifiers and interface seeding routines that enable the application of different material

Many physical problems can be modelled by partial differential equations (PDEs) on unknown domains. Several examples can easily be found in the dynamics of free interfaces in fluid dynamics, solid mechanics or in fluid-solid interactions. To solve these equations in an arbitrary domain with nonlinear deformations, the Arbitrary Lagrangian-Eulerian (ALE) method is in widespread use. For this method

We investigate a numerical method for approximating the solution of the one dimensional acoustic wave problem, when violating the numerical stability condition. We use deep learning to create an explicit non-linear scheme that remains stable for larger time steps and produces better accuracy than the reference implicit method. The proposed spatio-temporal neural-network architecture is additionally

High-order accurate kinetic energy and entropy preserving (KEEP) schemes in generalized curvilinear coordinates are proposed for stable and non-dissipative numerical simulations. The proposed schemes are developed on the basis of the physical relation that the fluxes in the Euler equations in generalized curvilinear coordinates can be derived by taking the inner product between the inviscid fluxes

We study the convergence of the discontinuous Galerkin (DG) method applied to the advection–reaction equation on meshes with reentrant faces. On such meshes, the upwind numerical flux is not smooth, and so the numerical integration of the resulting face terms can only be expected to be first-order accurate. Despite this inexact integration, we prove that the DG method converges with order O(hp+1/2)

A physically consistent approach is introduced to simulate dynamics of droplets in contact with solid substrates. The numerical method is developed by introducing the molecular–kinetic model within the framework of the level-set/enriched finite element method and including the theoretically resolved sub-elemental hydrodynamics. The level-set method is customized to comply fully with the model acquired

This paper presents a new numerical approach to the simulation of plasma equilibrium evolution in tokamaks. It is based on an unusual coupling formulation of current diffusion and axi-symmetric equilibrium equations that does not require the use of flux-surface averages and allows for fast Newton-type iterations schemes. A detailed derivation of the coupled system of non-linear equations and its discretization

Turbulent fluid flows in atmospheric and oceanic sciences are characterized by strongly transient features with spatial inhomogeneity, spanning a wide range of spatial and temporal scales. While large-scale dynamics are often well approximated by closure schemes there is still a need to efficiently represent the corresponding small-scale features, when it comes to the risk analysis for extreme events

In this paper, we construct and analyze a uniquely solvable, positivity preserving and unconditionally energy stable finite-difference scheme for the periodic three-component Macromolecular Microsphere Composite (MMC) hydrogels system, a ternary Cahn-Hilliard system with a Flory-Huggins-deGennes free energy potential. The proposed scheme is based on a convex-concave decomposition of the given energy

This work presents two unsplit geometric VOF schemes that extend the two-dimensional cellwise conservative unsplit (CCU) scheme [Comminal et al., J. Comput. Phys. 283 (2015) 582–608] to three dimensions. The novelty of the 3D-CCU schemes lies in the representation of the streaksurfaces of donating regions by polyhedral surfaces whose vertices are calculated with the 4th order Runge-Kutta scheme. Moreover

We present a fully Eulerian hybrid immersed-boundary/phase-field model to simulate wetting and contact line motion over any arbitrary geometry. The solid wall is described with a volume-penalisation ghost-cell immersed boundary whereas the interface between the two fluids by a diffuse-interface method. The contact line motion on the complex wall is prescribed via slip velocity in the momentum equation

Balsara's genuinely multidimensional Riemann solvers, which possess a simple closed-form and high efficiency in the low-order cases, realize their multidimensional effect successfully by introducing the vertex-based framework. However, traditional higher-order (third-order and above) reconstruction methods built for the structured grids become ambiguous when applied directly into these solvers. Nowadays

In this work, the original boundary condition-enforced immersed boundary method (IBM) [Wu and Shu (2009) [1], (2010) [2]] is improved to efficiently simulate incompressible flows with moving boundaries. The original boundary condition-enforced IBM can accurately interpret the no-slip boundary condition but becomes computationally tedious in simulating moving boundary problems due to the assembly of

Transcranial ultrasound therapy is increasingly used for the non-invasive treatment of brain disorders. However, conventional numerical wave solvers are currently too computationally expensive to be used online during treatments to predict the acoustic field passing through the skull (e.g., to account for subject-specific dose and targeting variations). As a step towards real-time predictions, in the

The least squares method with deep neural networks as function parametrization has been applied to solve certain high-dimensional partial differential equations (PDEs) successfully; however, its convergence is slow and might not be guaranteed even within a simple class of PDEs. To improve the convergence of the network-based least squares model, we introduce a novel self-paced learning framework, SelectNet

Immersed finite element methods are designed to solve interface problems on interface-unfitted meshes. However, most of the study, especially analysis, is mainly limited to the two-dimension case. In this paper, we provide an a priori analysis for the trilinear immersed finite element method to solve three-dimensional elliptic interface problems on Cartesian grids consisting of cuboids. We establish

This paper presents an extension of Discrete Fracture Matrix (DFM) models to compositional two-phase Darcy flow accounting for phase transitions and Fickian diffusion. The hybrid-dimensional model is based on nonlinear transmission conditions at matrix fracture (mf) interfaces designed to be consistent with the physical processes. They account in particular for the saturation jump induced by the different

In this work, we present a novel higher-order smooth artificial viscosity method for the discontinuous Galerkin spectral element method and related high order methods. A neural network is used to detect the need for stabilization. Inspired by techniques from image edge detection, the neural network locates discontinuities inside mesh elements on a sub-cell level. Once the sub-cell positions of the

In this study, a family of second-order process based modified Patankar Runge-Kutta schemes is proposed. The proposed schemes preserve both the mass and mole balance in a stiff reaction network while maintaining the positivity of density and pressure. The accuracy analysis is conducted to derive the sufficient and necessary conditions for the Runge-Kutta and Patankar coefficients. Coupling with the

By using the Onsager variational principle as an approximation tool, we develop a new diffusion generated motion method for wetting problems. The method uses a signed distance function to represent the interface between the liquid and vapor surface. In each iteration, a linear diffusion equation with a linear boundary condition is solved for one time step in addition to a simple re-distance step and

A new numerical method is developed for the robust simulation of high-Reynolds number two-fluid flows with high density contrast. The proposed method is a combination of the consistent mass–momentum transport framework of Nangia et al. (J. Comput. Phys., 2019, 390: 548–594) and the coupled level-set and volume-of-fluid method of Sussman and Puckett (J. Comput. Phys., 2000, 162: 301–337). The conservative

We propose an Edge Multiscale Finite Element Method (EMsFEM) based on an Interior Penalty Discontinuous Galerkin (IPDG) formulation for the heterogeneous Helmholtz problems with large wavenumber. A novel local multiscale space is constructed by solving local problems with a mixed boundary condition composed of a nonhomogeneous Dirichlet boundary condition and an absorbing boundary condition, which

In its application to the modeling of a mineral separation process, we propose the numerical analysis of the Cahn-Hilliard equation by employing space-time discretizations of the automatic variationally stable finite element (AVS-FE) method. The AVS-FE method is a Petrov-Galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous Petrov-Galerkin (DPG) method

The focus of this contribution is the numerical treatment of interface-coupled problems concerned with the interaction of impermeable or permeable elastic bodies1 and the surrounding fluid including the contact interaction of deformable bodies. The presented formulation is based on the approach for fluid-structural-contact interaction in [1], and consistently takes into account fluid-filled poroelastic

A refined model is presented which numerically resolves the fluid flow within a fracture inside a saturated poroelastic material. At the discontinuity, the mass balance of the fluid is solved using the velocity profile inside the fracture, with the velocity profile being determined numerically from the momentum balance in the integration points at the discontinuity. The resolution of the mass balance

We present a numerical method, based on a tensor order parameter description of a nematic phase, that allows fully anisotropic elasticity. Our method thus extends the Landau-de Gennes Q-tensor theory to anisotropic phases. A microscopic model of the nematogen is introduced (the Maier-Saupe potential in the case discussed in this paper), combined with a constraint on eigenvalue bounds of Q. This ensures

Element-wise techniques based on SVD or QR Decomposition completely get rid of the usual ill-conditioning inherent to the plane-wave discretizations of the Ultra-Weak Variational Formulation (UWVF) for the Helmholtz equation. Associated preconditioning strategies lead to very low condition numbers of the corresponding linear system matrices, without any limitation on the number of plane waves per element

We present scalable algorithms to simulate large-scale stochastic particle systems amenable for modeling dense colloidal suspensions, glasses and gels. To handle the large number of particles and consequent many-body interactions present in such systems, we leverage an Accelerated Stokesian Dynamics (ASD) approach, for which we developed parallel algorithms in a distributed memory architecture. We

One of the challenges when simulating astrophysical flows with self-gravity is to compute the gravitational forces. In contrast to the hyperbolic hydrodynamic equations, the gravity field is described by an elliptic Poisson equation. We present a purely hyperbolic approach by reformulating the elliptic problem into a hyperbolic diffusion problem, which is solved in pseudotime, using the same explicit

In this paper we present a methodology for increasing the accuracy and accelerating the convergence of numerical methods for solving Maxwell's equations in the frequency domain by taking into account the behavior of the electromagnetic field near the geometric edges of wedge-shaped structures. Several algorithms for incorporating treatment of singularities into methods for solving Maxwell's equations

This paper proposes and implements the Random Choice Method (RCM) based on a direct Eulerian generalized Riemann problem (GRP) scheme for one-dimensional Euler equations in gas dynamics. It is an application of the second order accurate GRP scheme proposed by Ben-Artzi et al. (2006) [6]. Since the RCM was introduced as a numerical tool in the gas dynamics and it has the advantage of capturing discontinuities

To numerically simulate the radiation diffusion problem with high parallel efficiency, we present a new parallel iteration algorithm for nonlinear diffusion equations. The algorithm is based on the domain decomposition method, and it integrates time extrapolation techniques and an advanced Jacobi explicit scheme. The domain decomposition method decomposes a large global problem into multiple sub-problems

We introduce a data-driven, linear method for the spatiotemporal prediction of high-dimensional and chaotic dynamical systems. In this method, the observables are vector-valued and delay-embedded, and the nonlinearities are treated as external forcings. The proper representation of the nonlinear terms is found in a physics-informed way or a purely data-driven fashion, depending on whether any knowledge

This paper extends the second-order accurate BGK finite volume schemes for the ultra-relativistic flow simulations [5] to the 1D and 2D special relativistic hydrodynamics with the Synge equation of state. It is shown that such 2D schemes are very time-consuming due to the moment integrals (triple integrals) so that they are no longer practical. In view of this, the simplified BGK (sBGK) schemes are

We consider the method-of-moments approach to solve the Boltzmann equation of rarefied gas dynamics, which results in the following moment-closure problem. Given a set of moments, find the underlying probability density function. The moment-closure problem has infinitely many solutions and requires an additional optimality criterion to single-out a unique solution. Motivated from a discontinuous Galerkin

Taking into account the limited accuracy of the energy-preserving exponential integrator of order two (Li and Wu, 2016 [29]) for conservative or dissipative systems with highly oscillatory solutions, this paper is devoted to presenting a uniform framework to design energy-preserving exponential integrators of arbitrarily high order based on the modifying integrator theory. To this end, we first show

This paper introduces a sharp interface method to simulate fluid-structure interaction (FSI) involving rigid bodies immersed in viscous incompressible fluids. The capabilities of this methodology are benchmarked using a range of test cases and demonstrated using large-scale models of biomedical FSI. The numerical approach developed herein, which we refer to as an immersed Lagrangian-Eulerian (ILE)

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

This paper presents a one-sided immersed boundary (IB) method using kernel functions constructed via a moving least squares (MLS) method. The resulting kernels effectively couple structural degrees of freedom to fluid variables on only one side of the fluid-structure interface. This reduces spurious feedback forcing and internal flows that are typically observed in IB models that use isotropic kernel

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

Sensitivity of periodic solutions of time-dependent partial differential equations is commonly computed using time-consuming direct and adjoint time integrations. Particular attention must be provided to the periodicity condition in order to obtain accurate results. Furthermore, stabilization techniques are required if the orbit is unstable. The present article aims to propose an alternative methodology

Subsurface geological formations provide giant capacities for large-scale (TWh) storage of renewable energy, once this energy (e.g. from solar and wind power plants) is converted to green gases, e.g. hydrogen. The critical aspects of developing this technology to full-scale will involve estimation of storage capacity, safety, and efficiency of a subsurface formation. Geological formations are often

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

In this work, an implicit scheme for particle-in-cell/Fourier electromagnetic simulations is developed and applied to studies of Alfvén waves in one dimension and three dimensional tokamak plasmas. An analytical treatment is introduced to achieve efficient convergence of the iterative solution of the implicit field-particle system. First, its application to the one-dimensional uniform plasma demonstrates

Various materials and solid-fluid mixtures of engineering and biomedical interest can be modelled as generalised Newtonian fluids, as their viscosity depends locally on the flow field. Despite the peculiarities of such models, it is common practice to combine them with numerical techniques conceived for Newtonian fluids, which can bring several issues such as spurious pressure boundary layers, unsuitable

High-order finite difference schemes are widely used in the field of numerical solutions of partial differential equations. Although the importance of their local conservation property is well recognized, especially for solving hyperbolic conservation laws, the importance of their global conservation property (GCP) is seldom considered. To show that in some cases the GCP is indeed important in improving

A subwavelength metallic particle supports localized surface plasmons for some negative permittivity values, which are eigenvalues of the self-adjoint quasi-static plasmonic eigenvalue problem (PEP). This work investigates the existence of complex plasmonic resonances for a 2D particle whose boundary is smooth except for one straight corner. These resonances are defined using the multivalued nature

A novel free–energy stable discontinuous Galerkin method is developed for the Cahn–Hilliard equation with non–conforming elements. This work focuses on dynamic polynomial adaptation (p–refinement) and constitutes an extension of the method developed by Manzanero et al. in Journal of Computational Physics 403:109072, 2020, which makes use of the summation–by–parts simultaneous–approximation term technique

Viscoelastic surface rheology plays an important role in multiphase systems. A typical example is the actin cortex which surrounds most animal cells. It shows elastic properties for short time scales and behaves viscous for longer time scales. Hence, realistic simulations of cell shape dynamics require a model capturing the entire elastic to viscous spectrum. However, currently there are no numerical

The paper is concerned with an adjoint complement to the Volume-of-Fluid (VoF) method for immiscible two-phase flows, e.g. air and water, which is widely used in marine engineering due to its computational efficiency. The particular challenge of the primal and the corresponding adjoint VoF-approach refers to the sharp interface treatment featuring discontinuous physical properties. Both the continuous

We propose a novel composite framework to find unknown fields in the context of inverse problems for partial differential equations (PDEs). We blend the high expressibility of deep neural networks as universal function estimators with the accuracy and reliability of existing numerical algorithms for partial differential equations as custom layers in semantic autoencoders. Our design brings together

This paper proves that shadowing solutions can be almost surely nonphysical. This finding invalidates the argument that small perturbations in a chaotic system can only have a small impact on its statistical behavior. This theoretical finding has implications for many applications in which chaotic dynamics plays an important role. It suggests, for example, that we can control the climate through subtle