Gaseous flows show a diverse set of behaviors on different characteristic scales. Given the coarse-grained modeling in theories of fluids, considerable uncertainties may exist between the flow-field solutions and the real physics. To study the emergence, propagation and evolution of uncertainties from molecular to hydrodynamic level poses great opportunities and challenges to develop both sound theories

The compressible Euler equations coupled with the gravitational source terms admit a hydrostatic equilibrium state where the gradients of the flux terms can be exactly balanced by those in the source terms. This property of exact preservation of the equilibrium is highly desirable at the discrete level when they are numerically solved. In this study, we design the simple high order well-balanced finite

We develop a multirate timestepper for semi-implicit solutions of the unsteady incompressible Navier-Stokes equations (INSE) based on a recently-developed multidomain spectral element method (SEM) [1]. For incompressible flows, multirate timestepping (MTS) is particularly challenging because of the tight coupling implied by the incompressibility constraint, which manifests as an elliptic subproblem

We present a new family of high-order shock-capturing finite difference numerical methods for systems of conservation laws. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered (2p+1)-point stencils, where p may take values in {1,2,…,P} according to a new family of smoothness indicators in the stencils. The methods are based on a combination of a robust first order

We propose an efficient method to reinitialize a level set function to a signed distance function by solving an elliptic problem using the finite element method. The original zero level set interface is preserved by means of applying modified boundary conditions to a surrogate/approximate interface weakly with a penalty method. Narrow band technique is adopted to reinforce the robustness of the proposed

We propose a new class of Bayesian neural networks (BNNs) that can be trained using noisy data of variable fidelity, and we apply them to learn function approximations as well as to solve inverse problems based on partial differential equations (PDEs). These multi-fidelity BNNs consist of three neural networks: The first is a fully connected neural network, which is trained following the maximum a

An alternating directional implicit (ADI)-Yee's scheme is developed for Maxwell's equations with discontinuous material coefficients along one or several interfaces. In order to use Yee's scheme with the presence of discontinuities, some intermediate quantities along the interface are introduced. The intermediate quantities are from the solutions and their derivatives on the interface and should satisfy

The goal of this work is to train a neural network which approximates solutions to the Navier-Stokes equations across a region of parameter space, in which the parameters define physical properties such as domain shape and boundary conditions. The contributions of this work are threefold: 1. To demonstrate that neural networks can be efficient aggregators of whole families of parameteric solutions

In this paper, we propose a procedure combining Lattice-Boltzmann and finite element simulations to model the effects of capillary pressure in porous microstructures. Starting from an explicit geometry of the microstructure, the Lattice-Boltzmann method is used to simulate the condensation from vapor phase to liquid and predict the geometry of capillary liquid films and liquid phases for arbitrary

In this paper, a domain decomposition technique in the finite volume framework is presented to propagate small amplitude acoustic and entropy waves in a linearized Euler region and simulate the interaction of these waves with an initially steady normal shock in a nonlinear region. An overset method is used to two-way couple the linear and nonlinear regions that overlap each other. Linearized solvers

We propose an accelerated density matrix purification scheme with error control. The method makes use of the scale-and-fold acceleration technique and screening of submatrix products in the block-sparse matrix-matrix multiplies to reduce the computational cost. An error bound and a parameter sweep are combined to select a threshold value for the screening, such that the error can be controlled. We

The leapfrog integrator is routinely used within the Hamiltonian Monte Carlo method and its variants. We give strong numerical evidence that alternative, easy to implement algorithms yield fewer rejections with a given computational effort. When the dimensionality of the target distribution is high, the number of accepted proposals may be multiplied by a factor of three or more. This increase in the

Tracking methods are used in particle transport Monte Carlo simulations to simulate the particle movements in models with multiple materials. However, as Monte Carlo simulations are becoming increasingly sophisticated, the most widely used tracking method, ray-tracing, exposes serious disadvantages in dealing with continuously varying materials and needs to be improved. Delta-tracking, the most common

The discontinuous Galerkin domain decomposition (DGDD) method couples subdomains of high-fidelity polynomial approximation to regions of low-dimensional resolution for the numerical solution of systems of conservation laws. In the low-fidelity regions, the solution is approximated by empirical modes constructed by Proper Orthogonal Decomposition and a reduced-order model is used to predict the solution

Rational exponential integrators (REXI) are a class of numerical methods that are well suited for the time integration of linear partial differential equations with imaginary eigenvalues. Since these methods can be parallelized in time (in addition to the spatial parallelization that is commonly performed) they are well suited to exploit modern high performance computing systems. In this paper, we

The paper is concerned with the three-dimensional electromagnetic scattering from a large open rectangular cavity that is embedded in a perfectly electrically conducting infinite ground plane. By introducing a transparent boundary condition, the scattering problem is formulated into a boundary value problem in the bounded cavity. Based on the Fourier expansions of the electric field, the Maxwell equation

In this paper, based on the Scalar Auxiliary Variable (SAV) approach [44], [45] and a newly proposed Lagrange multiplier (LagM) approach [22], [21] originally constructed for gradient flows, we propose two linear implicit pseudo-spectral schemes for simulating the dynamics of general nonlinear Schrödinger/Gross-Pitaevskii equations. Both schemes are of spectral/second-order accuracy in spatial/temporal

It is crucial to understand the degradation of implosion performance caused by three-dimensional physics and the implementation of 3-D spherical mesh is challenging. In this paper a mesh generation method based on the cube spherical projection is proposed for 3D implosion problem in Inertial Confinement Fusion (ICF). This kind of mesh has the advantage of good quality, convenient adding radiation source

We propose an effective and light-weight learning algorithm, Symplectic Taylor Neural Networks (Taylor-nets), to conduct continuous, long-term predictions of a complex Hamiltonian dynamic system based on sparse, short-term observations. At the heart of our algorithm is a novel neural network architecture consisting of two sub-networks. Both are embedded with terms in the form of Taylor series expansion

In this paper, we present a symmetric interior penalty discontinuous Galerkin finite element discretization for the numerical solution of second-order elliptic partial differential equations on general polygonal meshes using a reduced-space polygonal Bernstein-Bézier functions. They form a space of interpolatory functions (vice local polynomial functions), which satisfy the Lagrange property at their

A numerical method for TriGlobal (i.e., fully three-dimensional) adjoint stability analysis for compressible flows was developed and is presented in this paper. The developed method solves the adjoint stability problem using a matrix-free method based on Krylov-Schur method and a time-stepping approach. Because of the low memory (RAM) requirement of the matrix-free approach, the developed method can

This paper provides a detailed comparison in a solids mechanics context of adaptive mesh refinement methods for all-quadrilateral and all-hexahedral meshes. The adaptive multigrid Local Defect Correction method and the well-known hierarchical h-adaptive refinement techniques are placed into a generic algorithmic setting for an objective numerical comparison. Such a comparison is of great interest as

The solution of the pressure Poisson equation arising in the numerical solution of incompressible Navier–Stokes equations (INSE) is by far the most expensive part of the computational procedure, and often the major restricting factor for parallel implementations. Improvements in iterative linear solvers, e.g. deploying Krylov-based techniques and multigrid preconditioners, have been successfully applied

We propose a data-driven physics-informed finite-volume scheme for the approximation of small-scale dependent shocks. Nonlinear hyperbolic conservation laws with non-convex fluxes allow nonclassical shock wave solutions. In this work, we consider the cubic scalar conservation law as representative of such systems. As standard numerical schemes fail to approximate nonclassical shocks, schemes with controlled

We study in this paper three variants of the high-order Discontinuous Galerkin (DG) method with Runge-Kutta (RK) time integration for the induction equation, analysing their ability to preserve the divergence-free constraint of the magnetic field. To quantify divergence errors, we use a norm based on both a surface term, measuring global divergence errors, and a volume term, measuring local divergence

When describing the deflagration-to-detonation transition in solid granular explosives mixed with gaseous products of combustion, a well-developed two-phase mixture model is the compressible Baer-Nunziato (BN) model of flows containing solid and gas phases. As this model is numerically simulated by a conservative Godunov-type scheme, spurious oscillations are likely to generate from porosity interfaces

Due to its integro-differential nature, deriving schemes for numerically solving the radiative transfer equation (RTE) is challenging. Most solvers are efficient in specific scenarios: structured grids, simulations with low-scattering materials... In this paper, a full solver, from the discretization of the steady-state monochromatic RTE to the solution of the resulting system, is derived. Using a

We present a new method based on functional tensor decomposition and dynamic tensor approximation to compute the solution of a high-dimensional time-dependent nonlinear partial differential equation (PDE). The idea of dynamic approximation is to project the time derivative of the PDE solution onto the tangent space of a low-rank functional tensor manifold at each time. Such a projection can be computed

Numerical methods for the simulation of two-phase flows based on the common one-fluid model suffer from important transfer of momentum between the two-phases when the density ratio becomes important, such as with common air and water. This problem has been addressed from various numerical frameworks. It principally arises from the hypothesis that the momentum equation can be simplified by subtracting

In this paper we present a novel algorithm for simulating geometrical flows, and in particular the Willmore flow, with conservation of volume and area. The idea is to adapt the class of diffusion-redistanciation algorithms to the Willmore flow in both two and three dimensions. These algorithms rely on alternating diffusions of the signed distance function to the interface and a redistanciation step

We propose a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for an implementation of the Leray model that combines a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. The main novelty of the proposed approach relies in applying spatial filtering both for the collection of the snapshots and in the reduced order model,

A family of Time-Accurate and highly-Stable Explicit (TASE) operators for the numerical solution of stiff IVPs that includes those proposed by Bassenne et al. (2021) [1] is proposed. In this family the TASE operator of order k depends on k free parameters in contrast with Bassenne's family in which it depends only on one parameter to be chosen for stability and accuracy requirements. A complete study

The forward problems of pattern formation have been greatly empowered by extensive theoretical studies and simulations, however, the inverse problem is less well understood. It remains unclear how accurately one can use images of pattern formation to learn the functional forms of the nonlinear and nonlocal constitutive relations in the governing equation. We use PDE-constrained optimization to infer

We demonstrate how a multiplicative splitting method of order P can be utilized to construct an additive splitting method of order P+3. The weight coefficients of the additive method depend only on P, which must be an odd number. Specifically we discuss a fourth-order additive method, which is yielded by the Lie-Trotter splitting. We provide error estimates, stability analysis of a test problem, and

This paper discusses the development of algorithms for simulating compressible, four-way coupled, particle-laden flows with discontinuous Galerkin methods. Specifically, the algorithmic developments focus on the treatment of hard-sphere collisions. First, we aim to reduce the computational effort devoted to inspecting pairs of particles for collisions during the given time step. This is often one of

Multiphase flows are ubiquitous both in nature and industry. A broad range of interfacial scales, ranging from fine dispersions to large segregated interfaces, is often observed in such flows. Standard multiphase models rely on either the interface-averaging approach, which is suitable for the modelling of dispersed flows, or on the interface-resolving approach, which is ideal for large segregated

We present a novel interface-capturing scheme, THINC-scaling, to unify the VOF (volume of fluid) and the level set methods, which have been developed as two different approaches widely used in various applications. The key to success is to maintain a high-quality THINC reconstruction function using the level set field to accurately retrieve geometrical information and the VOF field to fulfill numerical

The Hamilton-Jacobi (HJ) equations arise in optimal control and many other applications. Oftentimes, such equations are posed in high dimensions, and this presents great numerical challenges. In this paper, we propose an adaptive sparse grid (also called adaptive multiresolution) local discontinuous Galerkin (DG) method for solving Hamilton-Jacobi equations in high dimensions. By using the sparse grid

Nonlinear radiative diffusion equation is coupled with fluid equations or magnetic hydrodynamical equations in radiation hydrodynamics simulations. The time scales of radiation and fluid can be different. To use the same time step in the coupled system, one needs to design numerical schemes for the nonlinear radiative diffusion equation that are stable for large time steps. Due to the nonlinearity

We introduce fast algorithms for generalized unnormalized optimal transport. To handle densities with different total mass, we consider a dynamic model, which mixes the Lp optimal transport with Lp distance. For p=1, we derive the corresponding L1 generalized unnormalized Kantorovich formula. We further show that the problem becomes a simple L1 minimization which is solved efficiently by a primal-dual

We construct finite element methods for the incompressible magnetohydrodynamics (MHD) system that precisely preserve the magnetic and cross helicity, the energy law and the magnetic Gauss law at the discrete level. The variables are discretized as discrete differential forms in a de Rham complex. We present numerical tests to show the performance of the algorithm.

A general purpose strategy is proposed to realize asymptotically the zero-variance importance sampling. The unknown integration constant can also be calculated simultaneously. This strategy can sample efficiently from multi-dimensional zero-variance importance function which is multi-modal by particular Markov Chain random walk. Sampling from this kind of distribution has been a challenge for a long

Deformable fractured porous media appear in many geoscience applications. While the extended finite element method (XFEM) has been successfully developed within the computational mechanics community for accurate modeling of deformation, its application in geoscientific applications is not straightforward. This is mainly due to the fact that subsurface formations are heterogeneous and span large length

To generate a mesh in a physical domain, an initial mesh of a polygonal domain that approximates the physical domain is introduced. The initial mesh is formed by using a Body Centered Cubic (BCC) lattice that can give a more efficient node ordering for the matrix vector multiplication. An optimization problem is then considered for the displacement on the initial mesh points, which maintains a good

Electroconvection is a multiphysics problem involving coupling of the flow field with the electric field as well as the cation and anion concentration fields. Here, we use electroconvection as a benchmark problem to put forward a new data assimilation framework, the DeepM&Mnet, for simulating multiphysics and multiscale problems at speeds much faster than standard numerical methods using pre-trained

In this paper, we present a new class of conservative semi-Lagrangian schemes for kinetic equations. They are based on the conservative reconstruction technique introduced in [1]. The methods are high order accurate both in space and time. Because of the semi-Lagrangian nature, the time step is not restricted by a CFL-type condition. Applications are presented to the Vlasov-Poisson system and the BGK

Deep neural networks (DNNs) are widely used as surrogate models, and incorporating theoretical guidance into DNNs has improved generalizability. However, most such approaches define the loss function based on the strong form of conservation laws (via partial differential equations, PDEs), which is subject to diminished accuracy when the PDE has high-order derivatives or the solution has strong discontinuities

This research gauges the ability of deep reinforcement learning (DRL) techniques to assist the control of conjugate heat transfer systems governed by the coupled Navier–Stokes and heat equations. It uses a novel, “degenerate” version of the proximal policy optimization (PPO) algorithm, intended for situations where the optimal policy to be learnt by a neural network does not depend on state, as is

In this paper we propose a new sampling-free approach to solve Bayesian model inversion problems that is an extension of the previously proposed spectral likelihood expansions (SLE) method. Our approach, called stochastic spectral likelihood embedding (SSLE), uses the recently presented stochastic spectral embedding (SSE) method for local spectral expansion refinement to approximate the likelihood

Presented in this work is a novel and easy-to-implement supplemental-frequency harmonic balance (SF-HB) approach to efficiently compute dynamically aperiodic systems, which can be cumbersome to model using the nominal high-dimensional harmonic balance (HDHB) technique. The stability of the time-spectral operator and the SF-HB solver involving multiple excitation frequencies is ensured by introducing

In this work, we uncover hidden geometric aspect of low-order compatible numerical schemes. First, we rewrite standard mimetic reconstruction operators defined by Stokes theorem using geometric elements of the barycentric dual grid, providing the equivalence between mimetic numerical schemes and discrete geometric approaches. Second, we introduce a novel global property of the reconstruction operators

Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to the computational domain Ω. One of the requirements leading to approximation robustness is to place the interpolation points with a locally uniform distance around the boundary of Ω. However generating interpolation points with such properties is a cumbersome problem. Instead

The paper describes a novel implementation of the piecewise linear interface-capturing volume-of-fluid method (PLIC-VOF) in axisymmetric cylindrical coordinates. The principal innovative feature involved in this work is that both the forward and inverse reconstruction problems are solved analytically, resulting in an appreciable speed-up in computing time in comparison with an iterative approach. All

This work attempts to solve two fundamental problems that can be encountered when solving kinetic equations using quadrature based moment methods (QBMM). These two problems are considered crucial problems since they are observed even in the simplest particulate systems. On the one hand, we have that particle trajectory crossing (PTC) events can lead to non-physical accumulation of particles in the

We propose an unsupervised machine-learning checkpoint-restart (CR) algorithm for particle-in-cell (PIC) algorithms using Gaussian mixtures (GM). The algorithm compresses the particle population per spatial cell by constructing a velocity distribution function using GM. Particles are reconstructed at restart time by local resampling of the Gaussians. To guarantee fidelity of the CR process, we ensure

We present an effective adaptive procedure for the numerical approximation of the steady-state Gross–Pitaevskii equation. Our approach is solely based on energy minimization, and consists of a combination of a novel adaptive finite element mesh refinement technique, which does not rely on any a posteriori error estimates, and a recently proposed new gradient flow. Numerical tests show that this strategy

Present finite-difference (FD) algorithms for modeling seismic wave propagation in 3D acoustic media are mainly based on the traditional orthogonal stencil and can achieve spatial high-order but temporal second-order accuracy. Therefore, these approaches may result in visible temporal dispersion and even instability for relatively large time sampling intervals. In this paper, we develop an advanced

A third-order gas-kinetic scheme (GKS) based on the subcell finite volume (SCFV) method is developed for the Euler and Navier-Stokes equations on triangular meshes, in which a computational cell is subdivided into four subcells. The scheme combines the compact high-order reconstruction of the SCFV method with the high-order flux evolution of the gas-kinetic solver. Different from the original SCFV

In this paper, we propose and analyze a positivity-preserving, energy stable numerical scheme for a certain type of reaction-diffusion systems involving the Law of Mass Action with the detailed balance condition. The numerical scheme is constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of reaction trajectories. The fact

We propose a new strategy for solving by the parareal algorithm highly oscillatory ordinary differential equations which are characteristics of a six-dimensional Vlasov equation. For the coarse solvers we use reduced models, obtained from the two-scale asymptotic expansions in [4]. Such reduced models have a low computational cost since they are free of high oscillations. The parareal method allows