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This paper focuses on a general computational framework to unify both Lagrangian staggered‐grid hydrodynamic (SGH) and cell‐centered hydrodynamic (CCH) methods. One challenge is that artificial viscosity has contained empirical parameters in the SGH method for seven decades. To address this challenge, a new relationship between pressure and velocity is constructed using specific volume as a medium. Another challenge is that entropy is increasing in isentropic flows for the CCH method. To overcome this second challenge, the forces acting on a target cell are split into linear and quadratic terms in the CCH method. The numerical results of the two methods are almost identical. The scheme is more general than both existing SGH and CCH methods. [ABSTRACT FROM AUTHOR]

This article presents a parameter uniform convergence numerical scheme for solving time fractional order singularly perturbed parabolic convection-diffusion differential equations with a delay. We give a priori bounds on the exact solution and its derivatives obtained through the problem’s asymp-totic analysis. The Euler’s method on a uniform mesh in the time direction and the extended cubic B-spline method with a fitted operator on a uniform mesh in the spatial direction is used to discretize the problem. The fitting factor is introduced for the term containing the singular perturbation pa-rameter, and it is obtained from the zeroth-order asymptotic expansion of the exact solution. The ordinary B-splines are extended into the extended B-splines. Utilizing the optimization technique, the value of μ (free param-eter, when the free parameter μ tends to zero the extended cubic B-spline reduced to convectional cubic B-spline functions) is determined. It is also demonstrated that this method is better than some existing methods in the literature.

In this paper, we propose a method for constructing a neural network viscosity in order to reduce the non-physical oscillations generated by high-order Discontinuous Galerkin methods on uniform Cartesian grids. To this end, the problem is reformulated as an optimal control problem for which the control is the viscosity function and the cost function involves comparison with a reference solution after several compositions of the scheme. The learning process is strongly based on gradient backpropagation tools. Numerical simulations show that the artificial viscosities, with a convolutional architecture, constructed in this way are just as good or better than those used in the literature. [ABSTRACT FROM AUTHOR]

Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this paper, we present a new viscous flux to regularize the MHD equations that holds many attractive properties. In particular, we prove that the proposed viscous flux preserves positivity of density and internal energy, satisfies the minimum entropy principle, is consistent with all generalized entropies, and is Galilean and rotationally invariant. We also provide a variation of the viscous flux that conserves angular momentum. To make the analysis more useful for numerical schemes, the divergence of the magnetic field is not assumed to be zero. Using continuous finite elements, we show several numerical experiments, including contact waves and magnetic reconnection. [ABSTRACT FROM AUTHOR]

This article presents a parameter uniform convergence numerical scheme for solving time fractional order singularly perturbed parabolic convectiondiffusion differential equations with a delay. We give a priori bounds on the exact solution and its derivatives obtained through the problem's asymptotic analysis. The Euler's method on a uniform mesh in the time direction and the extended cubic B-spline method with a fitted operator on a uniform mesh in the spatial direction is used to discretize the problem. The fitting factor is introduced for the term containing the singular perturbation parameter, and it is obtained from the zeroth-order asymptotic expansion of the exact solution. The ordinary B-splines are extended into the extended B-splines. Utilizing the optimization technique, the value of µ (free parameter, when the free parameter µ tends to zero the extended cubic B-spline reduced to convectional cubic B-spline functions) is determined. It is also demonstrated that this method is better than some existing methods in the literature. [ABSTRACT FROM AUTHOR]

The example of two families of finite-difference schemes shows that, in general, the numerical solution of the Riemann problem for the generalized Hopf equation depends on the finite-difference scheme. The numerical solution may differ both quantitatively and qualitatively. The reason for this is the nonuniqueness of the solution to the Riemann problem for the generalized Hopf equation. The numerical solution is unique in the case of a flow function with two inflection points if artificial dissipation and dispersion are introduced, i.e., the generalized Korteweg–de Vries-Burgers equation is considered. We propose a method for selecting coefficients of dissipation and dispersion. The method makes it possible to obtain a physically justified unique numerical solution. This solution is independent of the difference scheme. [ABSTRACT FROM AUTHOR]

The example of two families of finite-difference schemes shows that, in general, the numerical solution of the Riemann problem for the generalized Hopf equation depends on the finite-difference scheme. The numerical solution may differ both quantitatively and qualitatively. The reason for this is the nonuniqueness of the solution to the Riemann problem for the generalized Hopf equation. The numerical solution is unique in the case of a flow function with two inflection points if artificial dissipation and dispersion are introduced, i.e., the generalized Korteweg–de Vries-Burgers equation is considered. We propose a method for selecting coefficients of dissipation and dispersion. The method makes it possible to obtain a physically justified unique numerical solution. This solution is independent of the difference scheme.

This paper presents a parameter-uniform numerical method to solve the time dependent singularly perturbed delay parabolic convection-diffusion problems. The solution to these problems displays a parabolic boundary layer if the perturbation parameter approaches zero. The retarded argument of the delay term made to coincide with a mesh point and the resulting singularly perturbed delay parabolic convection-diffusion problem is approximated using the implicit Euler method in temporal direction and extended cubic B-spline collocation in spatial orientation by introducing artificial viscosity both on uniform mesh. The proposed method is shown to be parameter uniform convergent, unconditionally stable, and linear order of accuracy. Furthermore, the obtained numerical results agreed with the theoretical results.

This article presents an entropy-based corrective mechanism to improve nonlinear stability of computational algorithms in numerical heat transfer. The approach uses the transport form of the entropy production equation to calculate a parameter called the entropy-based artificial viscosity. A diffusion coefficient in the momentum conservation equations was modified based on the entropy-based artificial viscosity formulation. The corrective mechanism with an entropy-based artificial viscosity aims to utilize the Second Law as a stabilizing influence on erroneous numerical computations and enhance numerical stability and accuracy. Negative values of numerical entropy production due to discretization errors normally lead to physically unrealistic results that violate the numerical form of the Second Law. The algorithm uses these negative values as a predictive indicator to reduce numerical error and ensure closer compliance with the Second Law. The results for natural convection within a cavity indicate that the entropy-based artificial dissipation can significantly reduce the erroneous values of numerical entropy production and predicted velocities and temperatures, thereby improving the numerical accuracy and stability of the formulation. [ABSTRACT FROM AUTHOR]

In this paper, we recount the history of artificial viscosity, beginning with its origin in previously unpublished and unavailable documents, continuing on to current research and ending with recent work describing its physical basis that suggests new directions for improvement. This review is mainly about finite volume methods and the finite scale theory, We focus on the underlying ideas that recognize the finiteness of scale and of measurement. [ABSTRACT FROM AUTHOR]

Assuming Newton's law of cooling, the propagation and structure of isothermal acoustic shocks are studied under four different viscosity laws. Employing both analytical and numerical methods, 1D traveling wave solutions for the velocity and density fields are derived and analyzed. For each viscosity law considered, expressions for both the shock thickness and the asymmetry metric are determined. And, to ensure that isothermal flow is achievable, upper bounds on the associated Mach number values are derived/computed using the isothermal version of the energy equation. [ABSTRACT FROM AUTHOR]

The two-fluid model is widely used to describe two-phase flows in complex systems such as nuclear reactors. Although the two-phase flow was successfully simulated, the standard two-fluid model suffers from an ill-posed nature. There are several remedies for the ill-posedness of the one-dimensional (1D) two-fluid model; among those, artificial viscosity is the focus of this study. Some previous works added artificial diffusion terms to both mass and momentum equations to render the two-fluid model well-posed and demonstrated that this method provided a numerically converging model. However, they did not consider mass conservation, which is crucial for analyzing a closed reactor system. In fact, the total mass is not conserved in the previous models. This study improves the artificial viscosity model such that the 1D incompressible two-fluid model is well-posed, and the total mass is conserved. The water faucet and Kelvin-Helmholtz instability flows were simulated to test the effect of the proposed artificial viscosity model. The results indicate that the proposed artificial viscosity model effectively remedies the ill-posedness of the two-fluid model while maintaining a negligible total mass error.

The increasing demand for high-fidelity simulations of compressible turbulence on complex geometries poses a number of challenges for numerical schemes, and plenty of high-order methods have been developed. The high-order methods may encounter spurious oscillations or even blow up for strongly compressible flows, and a number of approaches have been developed, such as slope limiters and artificial viscosity models. In the family of artificial viscosity, which measures smoothness using the modal coefficients, the averaged modal decay (MDA) model employs all of the modes instead of only the highest mode as in the highest modal decay (MDH) model, which tends to underestimate the smoothness. However, the MDA approach requires high-order accuracy (usually P ≥ 4 ) to deliver a reliable estimation of smoothness. In this work, an approach used to extend the MDA model to lower orders, such as P 2 and P 3 , referred to as MDAEX, was proposed, where neighboring elements were incorporated to involve more information in the estimation process. A further controlling of the value of artificial viscosity was also introduced. The proposed model was applied to several typical benchmark cases and compared with other typical models. The results show that the MDAEX model recovers the expected accuracy better than the MDA model for P 2 and P 3 and captures flow structures well for shock-dominated flows. [ABSTRACT FROM AUTHOR]

In the simulation of compressible turbulent flows via a high-order flux reconstruction framework, the artificial viscosity model plays an important role to ensure robustness in the strongly compressible region. However, the impact of the artificial viscosity model in under-resolved regions on dissipation features or resolving ability remains unclear. In this work, the performance of a dilation-based (DB) artificial viscosity model to simulate under-resolved turbulent flows in a high-order flux reconstruction (FR) framework is investigated. Comparison is conducted with results via several typical explicit subgrid scale (SGS) models as well as implicit large eddy simulation (iLES) and their impact on important diagnostic quantities including turbulent kinetic energy, total dissipation rate of kinetic energy, and energy spectra are discussed. The dissipation rate of kinetic energy is decomposed into several components including those resulting from explicit SGS models or Laplacian artificial viscosity model; thus, an explicit evaluation of the dissipation rate led by those modeling terms is presented. The test cases consist of the Taylor-Green vortex (TGV) problem at R e = 1600 , the freely decaying homogeneous isotropic turbulence (HIT) at M a t 0 = 0.5 (the initial turbulent Mach number), the compressible TGV at Mach number 1.25 and the compressible channel flow at R e b = 15,334 (the bulk Reynolds number based on bulk density, bulk velocity and half-height of the channel), Mach number 1.5. The first two cases show that the DB model behaves similarly to the SGS models in terms of dissipation and has the potential to improve the insufficient dissipation of iLES with the fourth-order-accurate FR method. The last two cases further demonstrate the ability of the DB method on compresssible under-resolved turbulence and/or wall-bounded turbulence. The results of this work suggest the general suitability of the DB model to simulate under-resolved compressible turbulence in the high order flux reconstruction framework and also suggest some future work on controlling the potential excessive dissipation caused by the dilation term. [ABSTRACT FROM AUTHOR]

This paper describes a novel approach to alleviate the carbuncle phenomenon which consists in adding to any carbuncle prone Riemann solver an extra viscosity term in tangential momentum flux and its contribution to the energy conservation equation. This term contains one numerical parameter only, a scalar viscosity, which is reduced using a face-based shear detector to preserve shear waves. The idea stems from the investigation of some of the existing Riemann solvers, also presented in the paper. Indeed, when splitting the numerical flux into the face normal and tangential components, we observe that all the carbuncle free Riemann solvers present in the tangential part a numerical viscosity which scales with the sound speed when the normal flow velocity becomes zero. Opposite, in the carbuncle prone solvers this viscosity scales with the normal flow velocity. In particular the carbuncle free HLLCM scheme proposed by Shen et al. can be written by adding to the carbuncle prone HLLC scheme a tangential artificial viscosity term. Then the same can be done for any other Riemann solver, which renders the approach easy to implement in CFD codes for compressible flows. Numerical experiments shows the efficiency of the approach in computing carbuncle free single-component and multi-material flows. • Some Riemann solvers are investigated by splitting their numerical flux into interface normal and tangential components. • The carbuncle free ones present a tangential viscosity which scales with the sound speed as the normal velocity vanishes. • A tangential artificial viscosity approach is then proposed to alleviate the carbuncle problem to any Riemann solver. • The approach, combined with a shear sensor to preserve shear waves, is easy to implement in CFD codes. [ABSTRACT FROM AUTHOR]

This paper presents a shock capturing artificial viscosity scheme for the compact high-order finite volume methods in terms of the variational reconstructions on unstructured grids. The key for the design of the present artificial viscosity is the smoothness indicator, which is based on the concept of interfacial jump integration, measuring the discontinuities of the reconstruction polynomial and its spatial derivatives across a cell interface. Since the variational reconstruction is carried out by minimizing the functional in terms of the interfacial jump integration, the present smoothness indicator gives a discretization-consistent measurement of the smoothness of the flow fields that is sufficiently large in the region near discontinuities, and is in the same order of magnitude as the spatial truncation error of the finite volume scheme in smooth regions. These properties ensure that the newly developed artificial viscosity scheme has the problem-independent capability to suppress non-physical oscillations near discontinuities and preserve the theoretical order of accuracy for smooth flow. The shock capturing capability of the proposed artificial viscosity scheme has been demonstrated by a number of numerical examples confirming its essentially non-oscillatory and high-resolution properties. Additionally, the proposed artificial viscosity scheme exhibits higher computational efficiency than the approach based on a traditional limiter. [ABSTRACT FROM AUTHOR]

We present a novel high-order nodal artificial viscosity approach designed for solving Magnetohydrodynamics (MHD) equations. Unlike conventional methods, our approach eliminates the need for ad hoc parameters. The viscosity is mesh-dependent, yet explicit definition of the mesh size is unnecessary. Our method employs a multimesh strategy: the viscosity coefficient is constructed from a linear polynomial space constructed on the fine mesh, corresponding to the nodal values of the finite element approximation space. The residual of MHD is utilized to introduce high-order viscosity in a localized fashion near shocks and discontinuities. This approach is designed to precisely capture and resolve shocks. Then, high-order Runge-Kutta methods are employed to discretize the temporal domain. Through a comprehensive set of challenging test problems, we validate the robustness and high-order accuracy of our proposed approach for solving MHD equations. • New nodal-based artificial viscosity method for MHD. • The method does not include any ad hoc parameters or explicit definition of the mesh size. • The viscosity coefficient is built in a multigrid strategy. • The method is proven to preserve positivity for scalar conservation laws using linear finite elements. [ABSTRACT FROM AUTHOR]

The goal of this paper is to present an analytical solution, by means of the method of manufactured solutions (MMS), for the mathematical model that describes the behaviour of the grain mass aeration process, proposed by Thorpe. In contrast to related papers in the literature, several numerical approximations to solve the mathematical model were used. The finite difference method (FDM), employing the spatial approximations given by the methods of Roberts and Weiss, Leith, upwind difference scheme (UDS), central difference scheme (CDS) and UDS with deferred correction (UDS-C), combined with the explicit, implicit and Crank-Nicolson temporal formulations was applied. The effective order of the discretisation error achieved with the refinement of the mesh was verified by performing an error analysis for all approximations used. In addition, the results obtained numerically were compared to the analytical solution and the CPU (central processing unit) times at different levels of refinement. The difference in the CPU time using the methods CDS - Crank-Nicolson, Roberts and Weiss, and Leith, was very small compared to the method widely used in literature, the UDS - Explicit. It was also verified that the errors obtained by the proposed methods were considerably smaller than the error obtained by the UDS - Explicit method. In light of the above, the Leith method is recommended to numerically solve the grain mass aeration model proposed by Thorpe. • An analytical solution for a model that describes the aeration process was proposed. • Various numerical approximations to solve the model were studied. • Artificial viscosity was used to control oscillations in the studied model. • An error analysis on the numerical solution of the studied model was performed. [ABSTRACT FROM AUTHOR]

High order schemes have been investigated for quite a long time, and the flux reconstruction (FR) scheme proposed by Huynh recently attracts the attention of researchers due to its simplicity and efficiency. Building the framework that bridges discontinuous Galerkin (DG) and spectral difference (SD) schemes, FR recovers DG and SD conveniently with a careful selection of parameters. In this article, FR scheme is realized based on the framework of p4est, an open source adaptive mesh refinement library. The shock capturing ability of localized Laplacian artificial viscosity and in‐cell piecewise integrated solution methods are compared. Curved boundary treatment for high order schemes is adopted. The performance of developed code is estimated in both one and two dimensions including curved boundary and shock cases, and some attractive results are obtained. [ABSTRACT FROM AUTHOR]

We present a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method uses continuous Lagrange polynomials in space and explicit Runge-Kutta schemes in time. The shock-capturing term is based on the residual of MHD which tracks the shock and discontinuity positions, and adds sufficient amount of viscosity to stabilize them. The method is tested up to third order polynomial spaces and an expected fourth-order convergence rate is obtained for smooth problems. Several discontinuous benchmarks such as Orszag-Tang, MHD rotor, Brio-Wu problems are solved in one, two, and three spacial dimensions. Sharp shocks and discontinuity resolutions are obtained. [ABSTRACT FROM AUTHOR]

Abstract High order schemes have been investigated for quite a long time, and the flux reconstruction (FR) scheme proposed by Huynh recently attracts the attention of researchers due to its simplicity and efficiency. Building the framework that bridges discontinuous Galerkin (DG) and spectral difference (SD) schemes, FR recovers DG and SD conveniently with a careful selection of parameters. In this article, FR scheme is realized based on the framework of p4est, an open source adaptive mesh refinement library. The shock capturing ability of localized Laplacian artificial viscosity and in‐cell piecewise integrated solution methods are compared. Curved boundary treatment for high order schemes is adopted. The performance of developed code is estimated in both one and two dimensions including curved boundary and shock cases, and some attractive results are obtained.

This paper analyzes the sensitivity of artificial viscosity in the defect-deferred correction method for the non-stationary coupled Stokes/Darcy model. For the defect step and the deferred-correction step of the defect deferred correction method, we give the corresponding sensitivity systems related to the change of artificial viscosity. Finite element schemes are devised for computing numerical solutions to the sensitivity systems. Finally, we verify the theoretical analysis results through numerical experiments. The paper shows the effects of artificial viscosity, viscosity/hydraulic conductivity coefficients and spatial step sizes on sensitivity of numerical solutions to artificial viscosity in the defect step and the deferred correction step in detail. [ABSTRACT FROM AUTHOR]

We present an optimal transport approach for mesh adaptivity and shock capturing of compressible flows. Shock capturing is based on a viscosity regularization of the governing equations by introducing an artificial viscosity field as solution of the modified Helmholtz equation. Mesh adaptation is based on the optimal transport theory by formulating a mesh mapping as solution of Monge-Ampère equation. The marriage of optimal transport and viscosity regularization for compressible flows leads to a coupled system of the compressible Euler/Navier-Stokes equations, the Helmholtz equation, and the Monge-Ampère equation. We propose an iterative procedure to solve the coupled system in a sequential fashion using homotopy continuation to minimize the amount of artificial viscosity while enforcing positivity-preserving and smoothness constraints on the numerical solution. We explore various mesh monitor functions for computing r-adaptive meshes in order to reduce the amount of artificial dissipation and improve the accuracy of the numerical solution. The hybridizable discontinuous Galerkin method is used for the spatial discretization of the governing equations to obtain high-order accurate solutions. Extensive numerical results are presented to demonstrate the optimal transport approach on transonic, supersonic, hypersonic flows in two dimensions. The approach is found to yield accurate, sharp yet smooth solutions within a few mesh adaptation iterations. • An optimal transport approach is developed for shock capturing and mesh adaptation. • Minimize artificial viscosity subject to physicality and smoothness constraints. • Adapt meshes to capture shocks and resolve boundary layers. • Extensive results are presented for transonic, supersonic and hypersonic flows. [ABSTRACT FROM AUTHOR]

We present a novel shock-capturing strategy for high-order methods including discontinuous Galerkin (DG) method with data-driven gain optimization. Inspired by the classical control theory, we utilize a proportional–integral–derivative (PID) controller for capturing shock waves with monotonic subcell distributions. The proposed closed-loop control system for shock-capturing, named shock-capturing PID controller (SPID), consists of two key elements: error estimation based on the multi-dimensional limiting strategies (h MLP and h MLP_BD) and shock stabilization using the Laplacian artificial viscosity (LAV). The two elements are combined in a complementary manner to maximize the advantages of limiting strategy and artificial viscosity while overcoming each weakness. First, based on the multi-dimensional limiting process (MLP) condition and the troubled-boundary detector, the estimated error gives a signal to the SPID how much flow variables stray out of monotonic shock profiles. Second, the SPID estimates the amount of artificial viscosity to stabilize the target shock wave and superimposes numerical diffusion to the governing equations in the form of LAV. Each control action of the SPID (i.e., proportional, integral, and derivative) has a distinct role in capturing and stabilizing shock waves. The proportional action determines a minimal amount of background artificial viscosity. The integral action reinforces a shock-stabilizing numerical diffusion where the artificial viscosity by the proportional action alone is insufficient. The derivative action damps out sudden rises of error and spurious oscillations. The SPID incorporates the gain parameters that regulate the impact of each control action, and each gain parameter is determined via a surrogate-based optimization approach utilizing artificial neural networks (ANN). The SPID with the data-driven gain parameters is verified and validated by conducting extensive numerical tests and by comparing the results to other shock-capturing methods (h MLP, h MLP_BD, and Laplacian artificial viscosity). The numerical results demonstrate the excellent performance of SPID in terms of capturing shock waves and stabilizing shock-induced oscillations. Moreover, the SPID successfully preserves unsteady turbulent eddies for large-eddy simulations (LES) of subsonic and supersonic flows and improves convergence characteristics of steady transonic/supersonic flows. • A new shock-capturing method using a PID controller is proposed for high-order methods. • It combines advantages of limiter and artificial viscosity, while overcoming each weakness. • Accurate and smooth subcell resolution is achieved for shock–vortex interactions. • Eddy vortices in subsonic and supersonic turbulent simulations are preserved. • Convergence property is noticeably improved on steady supersonic/transonic flows. [ABSTRACT FROM AUTHOR]

The numerical oscillation problem is a difficulty for the simulation of rapidly varying shallow water surfaces which are often caused by the unsmooth uneven bottom, the moving wet-dry interface, and so on. In this paper, an adaptive artificial viscosity (AAV) is proposed and combined with the displacement shallow water wave equation (DSWWE) to establish an effective model which can accurately predict the evolution of multiple shocks effected by the uneven bottom and the wet-dry interface. The effectiveness of the proposed AAV is first illustrated by using the steady-state solution and the small perturbation analysis. Then, the action mechanism of the AAV on the shallow water waves with the uneven bottom is explained by using the Fourier theory. It is shown that the AVV can suppress the wave with the large wave number, and can also suppress the numerical oscillations for the rapidly varying bottom. Finally, four numerical examples are given, and the numerical results show that the DSWWE combined with the AAV can effectively simulate the shock waves, accurately capture the movements of wet-dry interfaces, and precisely preserve the mass. [ABSTRACT FROM AUTHOR]

The increasing demand for high-fidelity simulations of compressible turbulence on complex geometries poses a number of challenges for numerical schemes, and plenty of high-order methods have been developed. The high-order methods may encounter spurious oscillations or even blow up for strongly compressible flows, and a number of approaches have been developed, such as slope limiters and artificial viscosity models. In the family of artificial viscosity, which measures smoothness using the modal coefficients, the averaged modal decay (MDA) model employs all of the modes instead of only the highest mode as in the highest modal decay (MDH) model, which tends to underestimate the smoothness. However, the MDA approach requires high-order accuracy (usually P≥4) to deliver a reliable estimation of smoothness. In this work, an approach used to extend the MDA model to lower orders, such as P2 and P3, referred to as MDAEX, was proposed, where neighboring elements were incorporated to involve more information in the estimation process. A further controlling of the value of artificial viscosity was also introduced. The proposed model was applied to several typical benchmark cases and compared with other typical models. The results show that the MDAEX model recovers the expected accuracy better than the MDA model for P2 and P3 and captures flow structures well for shock-dominated flows.

A special modification of the combined scheme of the discontinuous Galerkin method, which increases the stability of this scheme when calculating discontinuous solutions with shock waves, is proposed. This modification is related to the addition of artificial viscosity of the fourth order of divergence to the basic scheme included in this combined scheme. The test calculations are presented that demonstrate the advantages of the new combined scheme in comparison with the standard monotonic versions of the discontinuous Galerkin method. [ABSTRACT FROM AUTHOR]

In the simulation of compressible turbulent flows via a high-order flux reconstruction framework, the artificial viscosity model plays an important role to ensure robustness in the strongly compressible region. However, the impact of the artificial viscosity model in under-resolved regions on dissipation features or resolving ability remains unclear. In this work, the performance of a dilation-based (DB) artificial viscosity model to simulate under-resolved turbulent flows in a high-order flux reconstruction (FR) framework is investigated. Comparison is conducted with results via several typical explicit subgrid scale (SGS) models as well as implicit large eddy simulation (iLES) and their impact on important diagnostic quantities including turbulent kinetic energy, total dissipation rate of kinetic energy, and energy spectra are discussed. The dissipation rate of kinetic energy is decomposed into several components including those resulting from explicit SGS models or Laplacian artificial viscosity model; thus, an explicit evaluation of the dissipation rate led by those modeling terms is presented. The test cases consist of the Taylor-Green vortex (TGV) problem at Re=1600, the freely decaying homogeneous isotropic turbulence (HIT) at Mat0=0.5 (the initial turbulent Mach number ), the compressible TGV at Mach number 1.25 and the compressible channel flow at Reb= 15,334 (the bulk Reynolds number based on bulk density, bulk velocity and half-height of the channel), Mach number 1.5. The first two cases show that the DB model behaves similarly to the SGS models in terms of dissipation and has the potential to improve the insufficient dissipation of iLES with the fourth-order-accurate FR method. The last two cases further demonstrate the ability of the DB method on compresssible under-resolved turbulence and/or wall-bounded turbulence. The results of this work suggest the general suitability of the DB model to simulate under-resolved compressible turbulence in the high order flux reconstruction framework and also suggest some future work on controlling the potential excessive dissipation caused by the dilation term.

We present a memory‐efficient high‐order hybridizable discontinuous Galerkin (HDG) formulation coupled with high‐order fully implicit Runge‐Kutta schemes for immiscible and incompressible two‐phase flow through porous media. To obtain the same high‐order accuracy in space and time, we propose using high‐order temporal schemes that allow using large time steps. Therefore, we require unconditionally stable temporal schemes for any combination of element size, polynomial degree, and time step. Specifically, we use the Radau IIA and Gauss‐Legendre schemes, which are unconditionally stable, achieve high‐order accuracy with few stages, and do not suffer order reduction in this problem. To reduce the memory footprint of coupling these spatial and temporal high‐order schemes, we rewrite the nonlinear system. In this way, we achieve a better sparsity pattern of the Jacobian matrix and less coupling between stages. Furthermore, we propose a fix‐point iterative method to further reduce the memory consumption. The saturation solution may present sharp fronts. Thus, the high‐order approximation may contain spurious oscillations. To reduce them, we introduce artificial viscosity. We detect the elements with high‐oscillations using a computationally efficient shock sensor obtained from the saturation solution and the post‐processed saturation of HDG. Finally, we present several examples to assess the capabilities of our formulation. [ABSTRACT FROM AUTHOR]

Two-dimensional dynamics of the wake in transitional flow around a circular cylinder is studied using weakly compressible smoothed particle hydrodynamics scheme up to 10 cylinder diameters downstream at a Reynolds number of 4000. The main objectives of this study are to evaluate the capability of SPH-LES and artificial viscosity in capturing flow activity as well as to test the performance of these formulations and background pressure coupling against the so-called tensile instability and to understand the effect of this instability in terms of mean integral quantities, first-order statistics and frequency spectra. It is observed that the effect of viscous formulation on flow dynamics becomes significant between four-and-seven diameters downstream of the cylinder. Background pressure coupling shifts the onset of transition about two diameters upstream by creating stiffer density field; however, it results in more energetic smaller scale activity. [ABSTRACT FROM AUTHOR]

When solving multidimensional problems of gas dynamics, finite-volume schemes using complete (i.e., based on a three-wave configuration) solvers of the Riemann problem suffer from shock-wave instability. It can appear as oscillations that cannot be damped by slope limiters, or it can lead to a qualitatively incorrect solution (carbuncle effect). To combat instability, one can switch to incomplete solvers based on a two-wave configuration near the shock wave, or introduce artificial viscosity. The article compares these two approaches on unstructured grids in relation to the EBR-WENO scheme for approximating convective terms and the classical Galerkin method for approximating diffusion terms. It is shown that the method of introducing artificial viscosity usually makes it possible to more accurately reproduce the flow pattern behind the shock front. However, on a three-dimensional unstructured grid, it causes dips ahead of the front, the depth of which depends on the quality of the grid, which can lead to an emergency stop of the calculation. Switching to an incomplete solver in this case gives satisfactory results with a much lower sensitivity to the quality of the mesh. [ABSTRACT FROM AUTHOR]

In this paper, we study numerical solutions for the Hamilton-Jacobi-Bellman (HJB) and Kolmogorov–Fokker–Planck (KFP) equations arising from mean field games. In order to solve the nonlinear discretized systems efficiently, we propose a multigrid method. Our proposed multigrid method is developed on the joint spacetime and is a full approximation scheme (FAS). We consider hybrid full-semi coarsening and kernel preserving biased restriction to address the anisotropy in time and convections in space. The main novelty of this paper is that we propose adding artificial viscosity to the direct discretization coarse grid operators, such that the coarse grid error estimations are more accurate. We use Fourier analysis to illustrate the efficiency of our proposed multigrid method. Numerical experiments show that the convergence rate of the proposed multigrid method is mesh-independent and faster than the existing methods in the literature. [ABSTRACT FROM AUTHOR]

A finite-difference algorithm is proposed for numerical modeling of hydrodynamic flows with rarefaction shocks, in which the fluid undergoes a jump-like liquid-gas phase transition. This new type of flow discontinuity, unexplored so far in computational fluid dynamics, arises in the approximation of phase-flip (PF) hydrodynamics, where a highly dynamic fluid is allowed to reach the innermost limit of metastability at the spinodal, upon which an instantaneous relaxation to the full phase equilibrium (EQ) is assumed. A new element in the proposed method is artificial kinetics of the phase transition, represented by an artificial relaxation term in the energy equation for a "hidden" component of the internal energy, temporarily withdrawn from the fluid at the moment of the PF transition. When combined with an appropriate variant of artificial viscosity in the Lagrangian framework, the latter ensures convergence to exact discontinuous solutions, which is demonstrated with several test cases. [ABSTRACT FROM AUTHOR]

A fully coupled high order discontinuous Galerkin (DG) solver for viscoelastic Oldroyd B fluid flow problems is presented. Contrary to known methods combining DG for the discretization of the convective terms of the material model with standard finite element methods (FEM) and using elastic viscous stress splitting (EVSS) and its derivatives, a local discontinuous Galerkin (LDG) formulation first described for hyperbolic convection-diffusion problems is used. The overall scheme is described, including temporal and spatial discretization as well as solution strategies for the nonlinear system, based on incremental increase of the Weissenberg number. The solvers suitability is demonstrated for the two-dimensional confined cylinder benchmark problem. The cylinder is immersed in a narrow channel with a blocking ratio of 1:2 and the drag force of is compared to results from the literature. Furthermore, steady and unsteady calculations give a brief insight into the characteristics of instabilities due to boundary layer phenomena caused by viscoelasticity arising in the narrowing between channel and cylinder. [ABSTRACT FROM AUTHOR]

Projection-based reduced order models rely on offline–online model decomposition, where the data-based energetic spatial basis is used in the expensive offline stage to obtain equations of reduced states that evolve in time during the inexpensive online stage. The online stage requires a solution method for the dynamic evolution of the coupled system of pressure and velocity states for incompressible fluid flows. The first contribution of this article is to demonstrate the applicability of the incremental pressure correction scheme for the dynamic evolution of pressure and velocity states. The evolution of a large number of these reduced states in the online stage can be expensive. In contrast, the accuracy significantly decreases if only a few reduced states are considered while not accounting for the interactions between unresolved and resolved states. The second contribution of this article is to compare three closure model forms based on global, modal and tensor artificial viscosity approximation to account for these interactions. The unknown model parameters are determined using two calibration techniques: least squares minimization of error in energy approximation and closure term approximation. This article demonstrates that an appropriate selection of solution methods and data-driven artificial viscosity closure models is essential for consistently accurate dynamics forecasting of incompressible fluid flows. [ABSTRACT FROM AUTHOR]

The conservative correction procedure of Abgrall [1] is studied from the perspective of filter-based artificial dissipation methods, which motivates the ability to tailor the behavior of the method in both physical and spectral space. Compared to the original formulation, employing diffusion operators biases the correction towards smaller scales and better controls discretization errors when seeking to enforce auxiliary conservation relations. Effective entropy-stable regularization of sharp gradients is furthermore shown to be attainable. Calculations of the Sod shock tube problem as governed by the one-dimensional Euler equations are used to highlight the utility of considering alternate filters within the original correction framework, where the notion of entropy conservation/stability is leveraged for improving non-linear scheme robustness. • Presentation of conservative correction procedure as specialized filter-based artificial dissipation (AD) scheme. • Motivation for correction to target erroneous modes in target auxiliary relations via choice of filter/AD stencil. • Perspective of equation-based partitionings of the correction. • Use of correction for entropy-stable regularization of sharp gradients by emulating a target artificial viscosity method. • Comparison of entropy preserving and stabilizing procedures for 1D Sod shock tube problem. [ABSTRACT FROM AUTHOR]

Focus is on the turbulence for a plunging breaker. Laser Doppler anemometer point measurements were used to determine the velocity matrix of a breaking wave on a sloping bottom. Using the Reynolds stress anisotropy for incompressible fluid, it was found that the ensemble averaged measured velocity predicted eddy viscosity is associated with peaks, which are absent in the broadly accepted empirical predictions. The instantaneous eddy viscosity coefficient was determined according to the Reynolds stresses, modified mean velocity and its gradient components and turbulent kinetic energy. The modified mean velocity and its derivatives improve eddy viscosity predictions during the wave period, which gives evidence that the velocity used corresponds well to a rotational part. In addition to the measurement predictions, empirical formulae were used to estimate the eddy viscosity values during the wave period. Furthermore, a meshless numerical model is proposed to determine artificial viscosity and demonstrate its dependence on eddy viscosity in the case of weakly compressible fluid.

Abstract Li et al. (SIAM J. Sci. Comput. 20:719–738, 1998) used the moving mesh partial differential equation (MMPDE) to solve a scaled Fisher’s equation and the initial condition consisting of an exponential function. The results obtained are not accurate because MMPDE is based on a familiar arc-length or curvature monitor function. Qiu and Sloan (J. Comput. Phys. 146:726–746, 1998) constructed a suitable monitor function called modified monitor function and used it with the moving mesh differential algebraic equation (MMDAE) method to solve the same problem of scaled Fisher’s equation and obtained better results. In this work, we use the forward in time central space (FTCS) scheme and the nonstandard finite difference (NSFD) scheme, and we find that the temporal step size must be very small to obtain accurate results. This causes the computational time to be long if the domain is large. We use two techniques to modify these two schemes either by introducing artificial viscosity or using the approach of Ruxun et al. (Int. J. Numer. Methods Fluids 31:523–533, 1999). These techniques are efficient and give accurate results with a larger temporal step size. We prove that these four methods are consistent for partial differential equations, and we also obtain the region of stability.