The immersed boundary method (IBM) with direct forcing is very popular in the simulation of rigid particulate flows. In the IBM, an interaction force is introduced at the interface between fluid and particle in order to approximate the no-slip boundary condition. The interaction force is calculated through dividing the velocity difference (or error) between fluid and particle at the interface by the

With the increasing use of high-order methods and high-order meshes, scientific visualization software need to adapt themselves to reliably render the associated meshes and numerical solutions. In this paper, a novel approach, based on OpenGL 4 framework, enables a GPU-based rendering of high-order meshes as well as an almost pixel-exact rendering of high-order solutions. Several aspects of the OpenGL

A stable spherical harmonics (PN) method for solving the problem of a multi-region sphere with internal sources subject to an externally incident angular flux is developed. The method consists in solving the problem locally (region by region) and using an iterative sweep technique to connect and update the solutions for the different regions until global convergence is attained. A post-processing step

Gene Expression Programming (GEP), a branch of machine learning, is based on the idea to iteratively improve a population of candidate solutions using an evolutionary process built on the survival-of-the-fittest concept. The GEP approach was initially applied with encouraging results to the modeling of the unclosed tensors in the context of RANS (Reynolds Averaged Navier–Stokes) turbulence modeling

A new method for mimetic interpolation on polygonal meshes is described. This new method is based on harmonic function interpolation. Explicit formulas for harmonic functions on general polygons do not exist, so truncated harmonic polynomial expansions are used for computational efficiency. We show that the naive harmonic polynomial expansion, is not stable for arbitrary polygons. However, higher level

We consider the numerical integration of moving boundary problems with the curve-shortening property, such as the mean curvature flow and Hele-Shaw flow. We propose a fully discrete curve-shortening polygonal evolution law. The proposed evolution law is fully implicit, and the key to the derivation is to devise the definitions of tangent and normal vectors and tangential and normal velocities at each

This work reports on the performances of a fully-discrete hp-adaptive entropy stable discontinuous collocated Galerkin method for the compressible Naiver–Stokes equations. The resulting code framework is denoted by SSDC, the first S for entropy, the second for stable, and DC for discontinuous collocated. The method is endowed with the summation-by-parts property, allows for arbitrary spatial and temporal

For the simulation of multiscale gas flows, the numerical scheme should be valid and efficient in both rarefied and continuum regimes. For example, the Direct Simulation Monte Carlo (DSMC) method is not appropriate because of its huge computational cost in the continuum regime. Various kinds of hybrid methods with DSMC have been proposed to deal with this difficulty. One of them is the particle-particle

Deep Learning research is advancing at a fantastic rate, and there is much to gain from transferring this knowledge to older fields like Computational Fluid Dynamics in practical engineering contexts. This work compares state-of-the-art methods that address uncertainty quantification in Deep Neural Networks, pushing forward the reduced-order modeling approach of Proper Orthogonal Decomposition-Neural

A new approach to model and compute the metric tensor corresponding to a-priori prescribed surface and volume boundary layers is presented. This is achieved by the use of sources that model the combined anisotropic metric. Algorithms are presented to both evaluate the resulting metric value at any point in space and the efficient approximation of the minimum and maximum metric over an axis-aligned

We develop an energy-stable scheme for simulating fluid-particle interaction problems governed by a coupled system consisting of the incompressible Navier-Stokes (NS) equations defined in a time-dependent fluid domain and Newton's second law for particle motion. A modified temporary arbitrary Lagrangian-Eulerian (tALE) method is designed based on a bijective mapping between the fluid regions at different

In this paper we present a mathematical and numerical model for human cardiac perfusion which accounts for the different length scales of the vessels in the coronary tree. Epicardial vessels are represented with fully three-dimensional (3D) fluid-dynamics, whereas intramural vessels are modeled as a multi-compartment porous medium. The coupling of these models takes place through interface conditions

In this study we have developed a flexible and efficient numerical scheme for the simulation of three–dimensional incompressible flows in spherical coordinates. The main idea, inspired by a similar strategy as [1] for cylindrical coordinates, consists of a change of variables combined with a discretization on a staggered mesh and the special treatment of few discrete terms that remove the singularities

We propose an efficient solver for saddle point problems arising from finite element approximations of nonlocal multi-phase Allen–Cahn variational inequalities. The solver is seen to behave mesh independently and to have only a very mild dependence on the number of phase field variables. In addition we prove convergence, in three GMRES iterations, of the approximation of the two phase problem, regardless

A high-order accurate finite-difference scheme modeling the fluid-structure interaction inside a pneumatic seismic source is presented. The model consists of two deforming fluid compartments separated by a moving shuttle. The fluid is governed by the 1D Euler equations. Well-posedness of the continuous problem is analyzed and proven in the frozen coefficient case. A stable discretization is derived

Flexible filaments and fibres are essential components of important complex fluids that appear in many biological and industrial settings. Direct simulations of these systems that capture the motion and deformation of many immersed filaments in suspension remain a formidable computational challenge due to the complex, coupled fluid–structure interactions of all filaments, the numerical stiffness associated

A classical reduced order model for dynamical problems involves spatial reduction of the problem size. However, temporal reduction accompanied by the spatial reduction can further reduce the problem size without losing much accuracy, which results in a considerably more speed-up than the spatial reduction only. Recently, a novel space–time reduced order model for dynamical problems has been developed

In this work we study the finite difference method for the fractional diffusion equation with high-dimensional hyper-singular integral fractional Laplacian. We first propose a simple and easy-to-implement discrete approximation, i.e., fractional centered difference scheme with γth-order (γ≤2) convergence for the fractional operator. Based on the established approximation, we then construct a finite

Unconditionally stable implicit time-marching methods are powerful in solving stiff differential equations efficiently. In this work, a novel framework to handle stiff physical terms implicitly is proposed. Both physical and numerical stiffness originating from convection, diffusion and source terms (typically related to reaction) can be handled by a set of predefined Time-Accurate and highly-Stable

Spectral methods solve elliptic partial differential equations (PDEs) numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time dependent problems, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. Spectral methods that converge spectrally in

In this article, we propose a novel high-order multi-moment finite volume method (MMFVM) for solving linear and nonlinear hyperbolic systems on unstructured grids. Different from the previous versions of volume-average/point-value multi-moment (VPM) scheme, the present scheme is developed by using a new reconstruction procedure based on the constrained least-square method which can be naturally extended

Summation-by-parts (SBP) finite difference methods have several desirable properties for second-order wave equations. They combine the computational efficiency of narrow-stencil finite difference operators with provable stability on curvilinear multiblock grids. While several techniques for boundary and interface conditions exist, weak imposition via simultaneous approximation terms (SATs) is perhaps

Elliptical PDEs are at the core of many computational problems. Sometimes it is necessary to solve them on fine meshes, which entail huge memory footprints and low computational speeds. We provide a method to solve elliptical PDEs on fine grids with lowered memory consumption and improved convergence. This paper considers one typical elliptical PDE – the Poisson equation on various polygonal domains

The Shifted Boundary Method (SBM) was recently introduced for the Poisson, linear advection/diffusion, Stokes, Navier-Stokes, acoustics, and shallow-water equations and belongs to the class of unfitted (or immersed, or embedded) finite element methods. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells

This paper presents a semi-implicit spectral deferred correction (SDC) method for incompressible Navier-Stokes problems with variable viscosity and time-dependent boundary conditions. The proposed method integrates elements of velocity- and pressure-correction schemes, which yields a simpler pressure handling and a smaller splitting error than the SDPC method of Minion and Saye (J. Comput. Phys. 375:

We consider a finite element method for Partial Differential Equations (PDEs) on surfaces. Unlike many previous techniques, this approach is based on a geometrically intrinsic formulation. With proper definition of the geometry and transport operators, the resulting finite element method is fully intrinsic to the surface. Here, we lay out in detail the formulation and compare it to a well-established

The stable and accurate approximation of discontinuities such as shocks on a finite computational mesh is a challenging task. Detection of shocks or strong discontinuities in the flow solution is typically achieved through a priori troubled cell indicators, which guide the subsequent action of an appropriate shock capturing mechanism. Arriving at a stable and accurate solution often requires empirically

In this paper, we develop a high order finite difference boundary treatment method for the implicit-explicit (IMEX) Runge-Kutta (RK) schemes solving hyperbolic systems with possibly stiff source terms on a Cartesian mesh. The main challenge is how to obtain the solutions at ghost points resulting from the wide stencil of the interior high order scheme. We address this problem by combining the idea

We present a detailed description and verification of a discontinuous Galerkin finite element method (DG) for the multi-component chemically reacting compressible Navier-Stokes equations that retains the desirable properties of DG, namely discrete conservation and high-order accuracy in smooth regions of the flow. Pressure equilibrium between adjacent elements is maintained through the consistent evaluation

A versatile and accurate treatment for the highly forward-peaked phase function in the three-dimensional (3D) radiative transfer equation (RTE) based on the discrete ordinates method (DOM) is crucial for biomedical optics. Our first objective was to compare the delta-Eddington (dE) and Galerkin quadrature (GQ) methods. The dE method decomposes the phase function into a purely forward-peaked component

An efficient separated method for fluid flows on collocated grid, named EPPL(Error Production and Propagation Limited), has been proposed involving robustness and convergence. The main features of the method include changing the solution procedure according to the matching degree of velocity and pressure dynamically, optimizing the under-relaxation strategy according to the characteristics of momentum

In this paper we propose a novel numerical approach for the Boltzmann equation with uncertainties. The method combines the efficiency of classical direct simulation Monte Carlo (DSMC) schemes in the phase space together with the accuracy of stochastic Galerkin (sG) methods in the random space. This hybrid formulation makes it possible to construct methods that preserve the main physical properties

In this paper we develop a new technique, called state redistribution, that allows the use of explicit time stepping when approximating solutions to hyperbolic conservation laws on embedded boundary grids. State redistribution is a postprocessing technique applied after each time step or stage of the base finite volume scheme, using a time step that is proportional to the volume of the full cells.

A framework is introduced that leverages known physics to reduce overfitting in machine learning for scientific applications. The partial differential equation (PDE) that expresses the physics is augmented with a neural network that uses available data to learn a description of the corresponding unknown or unrepresented physics. Training within this combined system corrects for missing, unknown, or

A hybrid scheme enabling multiphase simulations in complex flow setups with compressible flow solvers using a staggered arrangement of flow variables is developed. For this purpose, a Weighted Essentially Non-Oscillatory (WENO) scheme is adapted for staggered grids and combined with a finite difference central scheme. Hybrid weights maintaining continuity in time are computed based on a limiter constructed

Microfluidic “lab on a chip” devices are small devices that operate on small length scales on small volumes of fluid; these devices find uses in a variety of applications. Designs for microfluidic chips are generally composed of standardized and often repeated components connected by long, thin, straight fluid channels. We propose a novel meshing algorithm for use in simulating the linear incompressible

With rheology applications in mind, we present a fast solver for the time-dependent effective viscosity of an infinite lattice containing one or more neutrally buoyant smooth rigid particles per unit cell, in a two-dimensional Stokes fluid with given shear rate. At each time, the mobility problem is reformulated as a 2nd-kind boundary integral equation, then discretized to spectral accuracy by the

The discontinuous Galerkin (DG) method has been recognized as a promising approach for high-fidelity numerical simulations of turbulent flows (e.g., large-eddy simulation or direct numerical simulation), given its capability to achieve high-order accuracy in complex geometries while damping out spurious high-wavenumber oscillations in advection-dominated regimes. In this work, we analyze an advection-diffusion

Porous flow is of major importance in the shallow subsurface, since it directly impacts on reservoir-scale processes such as waste fluid sequestration or oil and gas exploration. Coupled and non-linear hydro-mechanical processes describe the motion of a low-viscous fluid interacting with a higher viscous porous rock matrix. This two-phase flow may trigger the initiation of solitary waves of porosity

While the potential groundbreaking role of mathematical modeling in electrophysiology has been demonstrated for therapies like cardiac resynchronization or catheter ablation, its extensive use in clinics is prevented by the need of an accurate customized conductivity identification. Data assimilation techniques are, in general, used to identify parameters that cannot be measured directly, especially

An immersed interface-lattice Boltzmann method (II-LBM) is developed for modeling fluid-structure systems. The key element of this approach is the determination of the jump conditions that are satisfied by the distribution functions within the framework of the lattice Boltzmann method where forces are imposed along a surface immersed in an incompressible fluid. In this initial II-LBM, the discontinuity

We introduce a fast Fourier spectral method for the spatially homogeneous Boltzmann equation with non-cutoff collision kernels. Such kernels contain non-integrable singularity in the deviation angle which arise in a wide range of interaction potentials (e.g., the inverse power law potentials). Albeit more physical, the non-cutoff kernels bring a lot of difficulties in both analysis and numerics, hence

We present well-balanced finite volume schemes designed to approximate the Euler equations with gravitation. They are based on a novel local steady state reconstruction. The schemes preserve a discrete equivalent of steady adiabatic flow, which includes non-hydrostatic equilibria. The proposed method works in Cartesian, cylindrical and spherical coordinates. The scheme is not tied to any specific numerical

This paper discusses the formulation of a class of hybrid compact schemes for structured irregular meshes. First, we derived three- and five-point compact schemes with required boundary closures for non-periodic problems. Second, we devised hybrid compact schemes from the explicit representation of the first- and second-order derivatives on irregular staggered and collocated meshes, respectively. For

This paper presents a numerical method for the simulation of fluid-structure interaction specifically tailored to interactions between Newtonian fluids and a large number of slender viscoelastic Cosserat rods. Because of their high flexibility and low weight the rods considered here exhibit large deflections, even under moderate fluid loads. Their motion, in turn, modifies the flow so that fluid and

We propose a set of quick and easy approximate characteristic variables for higher-order reconstructions of shock capturing schemes for magnetohydrodynamics (MHD). Numerical experiments suggest that the reconstructions using the approximate characteristic variables are more robust than those using the conservative or primitive variables, while their computational efficiencies are comparable. The approximate

A steady-state-preserving numerical model is developed for the shallow water equations in channels in the current study. The proposed numerical model is based on the finite volume method and capable to preserve both the moving-water and still-water steady states. In order to preserve the general steady states, the source terms in the momentum equation are incorporated into a global flux term; and the

The problem of mass diffusion in layered systems has relevance to applications in different scientific disciplines, e.g., chemistry, material science, soil science, and biomedical engineering. The mathematical challenge in these type of model systems is to match the solutions of the time-dependent diffusion equation in each layer, such that the boundary conditions at the interfaces between them are

In this paper, we propose a unified numerical framework for the time-dependent incompressible Navier–Stokes equation which yields the H1-, H(div)-conforming, and discontinuous Galerkin methods with the use of different viscous stress tensors and penalty terms for pressure robustness. Under minimum assumption on Galerkin spaces, the semi- and fully-discrete stability is proved when a family of implicit

The aim of this paper is to study the feasibility of time-reversal methods in a non homogeneous elastic medium, from data recorded in an acoustic medium. We aim to determine, from partial aperture boundary measurements, the presence and some physical properties of elastic unknown “inclusions”, i.e. not observable solid objects, located in the elastic medium. We first derive a variational formulation

We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. This new approach evolves the flow map using a combination of the Characteristic Mapping (CM) method [1] the gradient-augmented level-set (GALS) method [2]. The flow map possesses a semigroup structure which allows for the decomposition of a long-time deformation

We present a simple and effective method for evaluating double- and single-layer potentials for Laplace's equation in three dimensions close to the boundary. The close evaluation of these layer potentials is challenging because they are nearly singular integrals. The method we propose is based on writing these layer potentials in spherical coordinates where the point at which their kernels are peaked

The purpose of this paper is twofold. First, the application of high-order methods in combination with the popular HLLC Riemann solver demonstrates that the grid-aligned shock instability can strongly affect simulation results when the grid resolution is increased. Beyond the well-documented two-dimensional behavior, the problem is particularly troublesome with three-dimensional simulations. Hence

High-order velocity and pressure boundary conditions are presented in Eulerian incompressible smoothed particle hydrodynamics (ISPH). While the high-order convergence of Eulerian ISPH has been demonstrated by the authors for periodic internal flows using Gaussian kernels this was limited by first to second-order accuracy for cases with solid boundaries. Since the SPH interpolation method is numerically

In this paper, we construct and test a class of linear numerical schemes for the Functionalized Cahn-Hilliard (FCH) equation with a symmetric double-well potential function by using a stabilized scalar auxiliary variable (SAV) method. To get a fair assessment of these new SAV-type schemes, we compare output with numerical solutions obtained by the classical, fully-implicit BDF1 and BDF2 schemes. We

In this paper, we consider the Cahn-Hilliard equation coupled with the incompressible Navier-Stokes equation, usually known as the Cahn-Hilliard-Navier-Stokes (CHNS) system. The CHNS system has been widely embraced to investigate the dynamics of a binary fluid mixture. By utilizing the modified leap-frog time-marching method, we propose a novel numerical algorithm for solving the CHNS system in an

Immersed Interface Methods (IIM) arise as a very effective tool to solve many interface problems encountered in fluid dynamics, mechanics and other related fields of study. Despite their versatility and potential, IIM-inspired techniques impose as constraints different types of jump conditions in order to be mathematically tractable and usable in practice. To cope with this issue, in this paper we

We investigate the continuum limit that the number of beads goes to infinity in the ring polymer representation of thermal averages. Studying the continuum limit of the trajectory sampling equation sheds light on possible preconditioning techniques for sampling ring polymer configurations with large number of beads. We propose two preconditioned Langevin sampling dynamics, which are shown to have improved

In this paper, two modifications are introduced for improving the accuracy, versatility, and robustness of a class of hybrid methods for radiation transport. In general, such methods are constructed by splitting the radiative flux into collided and uncollided components to which low- and high-resolution angular approximations are applied, respectively. In this work we focus on discrete ordinates discretizations