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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 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]

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]

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]

We introduce a novel displacement-based material point method for simulating weakly compressible free-surface flows and fluid–structure interaction. To address volumetric locking, we employ a B¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{{\textbf {B}}}$$\end{document}/F¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{{\textbf {F}}}$$\end{document}-inspired technique, previously developed for solid mechanics. This technique involves projecting the pressure and the dilatational part of the velocity gradient onto a lower-dimensional approximation space, eliminating complexities associated with two-field mixed formulations and operator splitting approaches. Additionally, to mitigate spurious pressure oscillations resulting from the use of a density-dependent equation of state, we enhance the framework with an artificial viscosity term. Finally, we employ higher-order spline background shape functions, resulting in a continuous representation of the velocity gradient and effectively preventing pressure jumps when material points cross element boundaries. Challenging numerical examples are provided to verify and validate our approach, demonstrating results that closely align with existing literature, exhibit reduced pressure oscillations, and are free of volumetric locking issues. [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]

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]

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]

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]

High-order discretizations have become increasingly popular across a wide range of applications, including reservoir simulation. However, the lack of stability and robustness of these discretizations for advection-dominant problems prevent them from being widely adopted. This paper presents work towards improving the stability and robustness of the discontinuous Galerkin (DG) finite element scheme, for advection-dominant two-phase flow problems in particular. A linearized analysis of the two-phase flow equations is used to show that a standard DG discretization of the two-phase flow equations in mass conservation form results in a neutrally stable semi-discrete system in the advection-dominant limit. Furthermore, the analysis is also used to propose additional terms to the DG method which linearly stabilize the discretization. These additional terms are derived by comparing the linearized equations in mass conservation form against an upwinded pressure-saturation form of the equations. Next, a partial differential equation-based artificial viscosity method is proposed for the Buckley-Leverett and two-phase flow equations, as a means of mitigating Gibbs oscillations in high-order discretizations and ensuring convergence to physical solutions. The modified DG method with artificial viscosity is demonstrated on a two-phase flow problem with heterogeneous rock permeabilities, where the high-order discretizations significantly outperform a conventional first-order approach in terms of computational cost required to achieve a given level of error in an output of interest. [ABSTRACT FROM AUTHOR]

In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for scalar hyperbolic conservation laws in multidimensions. Compared with previous work for linear hyperbolic equations [W. Guo and Y. Cheng, SIAM J. Sci. Comput., 38 (2016), pp. A3381-A3409, W. Guo and Y. Cheng, SIAM J. Sci. Comput., 39 (2017), pp. A2962-A2992], a class of interpolatory multiwavelets are applied to efficiently compute the nonlinear integrals over elements and edges in DG schemes. The resulting algorithm, therefore, can achieve similar computational complexity as the sparse grid DG method for smooth solutions. Theoretical and numerical studies are performed taking into consideration the accuracy and stability with regard to the choice of the interpolatory multiwavelets. Artificial viscosity is added to capture the shock and only acts on the leaf elements taking advantage of the multiresolution representation. Adaptivity is realized by auto error thresholding based on hierarchical surplus. Accuracy and robustness are demonstrated by several numerical tests. [ABSTRACT FROM AUTHOR]

This report presents defect correcting extrapolation technique (DCE), an alternative approach for defect correction (DC). In contrast to conventional DC methods, which always solve for separate correction steps, DCE instantly assumes weighted linear combinations of several artificial viscosity approximations as a corrected solution. This way, it merely requires solving an identical problem with varying viscosities. The theory and computations show that DCE uncovers "hidden turbulence" in artificially-diffused laminar solutions, and recovers their energy and turbulent characteristics. [ABSTRACT FROM AUTHOR]

Stability is an important aspect of numerical methods for hyperbolic conservation laws and has received much interest. However, continuity in time is often assumed and only semidiscrete stability is studied. Thus, it is interesting to investigate the inuence of explicit and implicit time integration methods on the stability of numerical schemes. If an explicit time integration method is applied, spacially stable numerical schemes for hyperbolic conservation laws can result in unstable fully discrete schemes. Focusing on the explicit Euler method (and convex combinations thereof), undesired terms in the energy balance trigger this phenomenon and introduce an erroneous growth of the energy over time. In this work, we study the inuence of artificial dissipation and modal filtering in the context of discontinuous spectral element methods to remedy these issues. In particular, lower bounds on the strength of both artificial dissipation and modal filtering operators are given and an adaptive procedure to conserve the (discrete) L2 norm of the numerical solution in time is derived. This might be beneficial in regions where the solution is smooth and for long time simulations. Moreover, this approach is used to study the connections between explicit and implicit time integration methods and the associated energy production. By adjusting the adaptive procedure, we demonstrate that filtering in explicit time integration methods is able to mimic the dissipative behavior inherent in implicit time integration methods. This contribution leads to a better understanding of existing algorithms and numerical techniques, in particular the application of artificial dissipation as well as modal filtering in the context of numerical methods for hyperbolic conservation laws together with the selection of explicit or implicit time integration methods. [ABSTRACT FROM AUTHOR]

In this work, two numerical schemes were developed to overcome the problem of shock waves that appear in the solutions of one/two-layer shallow water models. The proposed numerical schemes were based on the method of lines and artificial viscosity concept. The robustness and efficiency of the proposed schemes are validated on many applications such as dam-break problem and the problem of interface propagation of two-layer shallow water model. The von Neumann stability of proposed schemes is studied and hence, the sufficient condition for stability is deduced. The results were presented graphically. The verification of the obtained results is achieved by comparing them with exact solutions or another numerical solutions founded in literature. The results are satisfactory and in much have a close agreement with existing results. [ABSTRACT FROM AUTHOR]

In general, B-spline quasi-interpolation (BSQI)-based numerical schemes for hyperbolic conservation laws are unstable in nature. In the present work, we have developed the stable modified version of the cubic B-spline quasi-interpolation (CBSQI) numerical scheme for the hyperbolic conservation laws in one space dimension. In order to stabilize the CBSQI scheme, we have added an adaptive artificial viscosity term and derived the conditions under which the modified scheme is monotone. Although the CBSQI scheme can be of at most fourth order accurate in space, the monotone condition restricts the resulting scheme to be of at most order of one, as expected. In order to improve the order of accuracy, we adopt a procedure, which we call the weighted adaptive viscosity approach, through which we achieved the fourth order accuracy both in space and time in the modified CBSQI scheme. Further, linear L 2 -stability of the modified CBSQI scheme is analyzed using von-Neumann analysis. Numerical experiments are performed to validate the efficiency and stability of the modified CBSQI scheme. [ABSTRACT FROM AUTHOR]

The investigation of impact biomechanics is of extremeimportance in the understanding of damageswhich are caused by a projectile impacting the human body. In a high-velocity impact framework, nonpenetrating impacts (nonlethal projectiles) or penetrating impacts can both lead to severe body trauma. An interesting way to study these phenomena is the numerical simulation, which can typically bring new information that cannot be accessed with experimental tests. Classical finite element is often used to simulate these kinds of complex problems. However, for perforating impacts, this method can rapidly lead to its limits, especially in terms of grid deterioration, when the projectile goes through thematerial. An alternative to this problem is to use a meshlessmethod, like Smoothed Particles Hydrodynamics (SPH), which does not involve any mesh of the structure. Thus, the present study proposes to investigate perforating impact in a ballistic gelatin material, which is known as a good biological tissue simulant, using an axisymmetrical SPH code. Axisymmetrical formulation of the problem is developed, using appropriate numerical features such as contact procedure (between the projectile and the material), corrective factors. Experimental tests of the literature involving different impact velocities and different projectile sizes are replicated with the developed code, showing very promising results in terms of penetration and velocity of the projectile in the target. [ABSTRACT FROM AUTHOR]

The present study aims to develop a parameter-free staggered-grid Lagrangian scheme that avoids the empirical parameters needed for artificial viscosity and anti-hourglass force. The artificial viscosity is equal to the pressure jump, which is constructed by the relationship between pressure and velocity. The anti-hourglass force is established using pressure compensation, which is the difference in pressure between sub-cells belonging to the same primary cell. The scheme maintains the conservation of total mass, momentum, and energy. Numerical experiments show that the scheme is highly robust for extreme flow problems with different fine meshes such as Sedov, Noh, Saltzmann triple point and Rayleigh–Taylor instability. The robust parameter-free scheme is well suited for multi-physics problems and engineering applications. • A parameter-free staggered-grid Lagrangian scheme is developed for 2D compressible flow problems. • A new form of artificial viscosity is constructed. • A new strategy for controlling the hourglass motion is presented. • The new scheme maintains the conservation of total mass, monument, and energy. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a high-order accurate finite element method for incompressible variable density flow. The method uses high-order Taylor-Hood velocity-pressure elements in space and backward differentiation formula (BDF) time stepping in time. This way of discretization leads to two main issues: (i) a saddle point system that needs to be solved at each time step; (i i) a stability issue when the viscosity of the flow goes to zero or if the density profile has a discontinuity. We address the first issue by using Schur complement preconditioning and artificial compressibility approaches. We observed similar performance between these two approaches. To address the second issue, we introduce a modified artificial Guermond-Popov viscous flux where the viscosity coefficients are constructed using a newly developed residual-based shock-capturing method. Numerical validations confirm high-order accuracy for smooth problems and accurately resolved discontinuities for problems in 2D and 3D with varying density ratios. • A robust and highly accurate finite element method for incompressible variable density flow is presented. • A modified Schur complement preconditioning is proposed that has comparable performance to artificial compressibility methods. • The proposed viscous regularization is kinetic energy stable. • A newly developed residual-based shock-capturing method is proposed. • High-order accuracy for smooth problems and accurately resolved discontinuities for problems with varying density ratios. [ABSTRACT FROM AUTHOR]

Goal-oriented mesh adaptation, in particular using the dual-weighted residual (DWR) method, is known in many cases to produce very efficient meshes. For obtaining such meshes the (numerical) solution of an adjoint problem is needed to weight the residuals appropriately with respect to their relevance for the overall error. For hyperbolic problems already the weak primal problem requires in general an additional entropy condition to assert uniqueness of solutions; this difficulty is also reflected when considering adjoints to hyperbolic problems involving discontinuities where again an additional requirement (reversibility) is needed to select appropriate solutions. Within this article, an approach to the DWR method for hyperbolic problems based on an artificial viscosity approximation is proposed. It is discussed why the proposed method provides a well-posed dual problem, while a direct, formal, application of the dual problem does not. Moreover, we will discuss a further, novel, approach in which the forward problem need not be modified, thus allowing for an unchanged forward solution. The latter procedure introduces an additional residual term in the error estimation, accounting for the inconsistency between primal and dual problem. Finally, the effectivity of the extended error estimator, assessing the global error by a suitable functional of interest, is tested numerically; and the advantage over a formal estimator approach is demonstrated. [ABSTRACT FROM AUTHOR]

The collective behavior of animals can be modeled by a system of equations of continuum mechanics endowed with extra terms describing repulsive and attractive forces between the individuals. This system can be viewed as a generalization of the compressible Euler equations with all of its unpleasant consequences, e.g., the non-uniqueness of solutions. In this paper, we analyze the equations describing a viscous approximation of a generalized compressible Euler system and we show that its dissipative measure-valued solutions tend to a strong solution of the Euler system as viscosity tends to zero, provided the strong solution exists. [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 FROM AUTHOR]

This is the second part to our companion paper [18]. Herein, we generalize to two space dimensions the C -method developed in [20,18] for adding localized, space-time smooth artificial viscosity to nonlinear systems of conservation laws that propagate shock waves, rarefaction waves, and contact discontinuities. For gas dynamics, the C -method couples the Euler equations to scalar reaction-diffusion equations, which we call C -equations, whose solutions serve as space-time smooth artificial viscosity indicators for shocks and contacts. We develop a high-order numerical algorithm for gas dynamics in 2- D which can accurately simulate the Rayleigh-Taylor (RT) instability with Kelvin-Helmholtz (KH) roll-up of the contact discontinuity, as well as shock collision and bounce-back. Solutions to our C -equations not only indicate the location of the shocks and contacts, but also track the geometry of the evolving fronts. This allows us to implement both directionally isotropic and anisotropic artificial viscosity schemes, the latter adding diffusion only in directions tangential to the evolving front. We additionally produce a novel shock collision indicator function, which naturally activates during shock collision, and then smoothly deactivates. Moreover, we implement a high-frequency 2- D wavelet-based noise detector together with an efficient and localized noise removal algorithm. To test the methodology, we use a highly simplified WENO-based discretization scheme. We provide numerical results for some classical 2- D test problems, including the RT problem, the Noh problem, a circular explosion problem from the Liska & Wendroff [13] review paper, the Sedov blast wave problem, the double Mach 10 reflection test, and a shock-wall collision problem. In particular, we show that our artificial viscosity method can eliminate the wall-heating phenomenon for the Noh problem, and thereby produce an accurate, non-oscillatory solution, even though our simplified WENO-type scheme fails to run for this problem. • A novel 2-D space-time smooth artificial viscosity method is developed. • Both isotropic and anisotropic directional viscosity with front tracking. • Stabilization of shock-wall and shock-contact collision. • 2-D wavelet-based noise detection and removal scheme. • Easy to implement, high-order numerical schemes. [ABSTRACT FROM AUTHOR]

• An space-time smooth artificial viscosity method for simulating shocks and contacts in 1-D gas dynamics is presented. • Our method is applied to the highly singular phenomenon of shock-wall collision. • A wavelet-based noise detection and removal scheme is developed. • The new algorithm is extremely simple to implement and produces highly accurate solutions. In this first part of two papers, we extend the C -method developed in [40] for adding localized, space-time smooth artificial viscosity to nonlinear systems of conservation laws that propagate shock waves, rarefaction waves, and contact discontinuities in one space dimension. For gas dynamics, the C -method couples the Euler equations to a scalar reaction-diffusion equation, whose solution C serves as a space-time smooth artificial viscosity indicator. The purpose of this paper is the development of a high-order numerical algorithm for shock-wall collision and bounce-back. Specifically, we generalize the original C -method by adding a new collision indicator, which naturally activates during shock-wall collision. Additionally, we implement a new high-frequency wavelet-based noise detector together with an efficient and localized noise removal algorithm. To test the methodology, we use a highly simplified WENO-based discretization scheme. We show that our scheme improves the order of accuracy of our WENO algorithm, handles extremely strong discontinuities (ranging up to nine orders of magnitude), allows for shock collision and bounce back, and removes high frequency noise. The causes of the well-known "wall heating" phenomenon are discussed, and we demonstrate that this particular pathology can be effectively treated in the framework of the C -method. This method is generalized to two space dimensions in the second part of this work [41]. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new multiphase SPH model with lower diffusion is developed for simulation of multiphase flows with large density ratios, complex interfaces, and strong impact. The zeroth-order density smoothing algorithm in previous models is replaced by a first-order density correction term which could effectively suppress density/pressure oscillation while lowering numerical diffusion. A switch-function-based artificial viscosity term is proposed to change the global implementation of numerical diffusion in the momentum equation into the local effectiveness depending on the strong impact. Such strategy aims to offer a good balance between the necessary numerical diffusion and the true physical viscosity of the fluid. In order to prove the robustness of the algorithm in various flow problems, five representative numerical examples, including Rayleigh–Taylor instability, internal solitary waves on the pycnocline, air bubble rising in water, oscillation of elliptic droplet and dam breaking, are presented. The expected advantages of the proposed method are demonstrated through comparison with the experimental data and previous numerical results. [ABSTRACT FROM AUTHOR]

This paper reports the numerical analysis of shock wave diffraction over a convex sharp splitter geometry, focusing on the mechanism of the shock diffraction and the longtime behavior of shock-vortex dynamics. The flow evolution with shock-vortex dynamics for incident shock Mach number, Ms=1.59, is found to be in excellent agreement with the previous experimental results. We use a recent entropy-generation-based artificial viscosity (AV) method in conjunction with a high-order explicit discontinuous spectral element method (DSEM) to resolve these complex interactions. The AV is coupled with a shock sensor switch to attain optimal dissipations. Simulations capture the essential wave diffraction, transverse wave interaction with the deforming and growing primary vortex, and weaker secondary vortices arising from the Kelvin-Helmholtz instability. A quantification of the artificial dissipation of the numerical scheme is made by comparing the components of the kinetic energy dissipation rate and the pressure dilatation term. A new detailed transient flow analysis is also presented to address the shock dynamics, shock-vortex interaction, and the evolution of the flow topology with the probability density functions of various parameters of the enstrophy transport equation and the invariants of the velocity gradient tensor. The analysis reveals the mechanism of unwinding of vortices and its link with the divergence of the Lamb vector. A positive correlation is found between enstrophy and the imaginary part of the eigenvalues. Real parts of the two eigenvalues are associated with high dilatation shock regions and the outer edges of the vortices, respectively. [ABSTRACT FROM AUTHOR]

Artificial viscosity is used in the computer simulation of high Reynolds number flows and is one of the oldest numerical artifices. In this paper, I will describe the origin and the interpretation of artificial viscosity as a physical phenomenon. The basis of this interpretation is the finite scale theory, which describes the evolution of integral averages of the fluid solution over finite (length) scales. I will outline the derivation of finite scale Navier-Stokes equations and highlight the particular properties of the equations that depend on the finite scales. Those properties include enslavement, inviscid dissipation, and a law concerning the partition of total flux of conserved quantities into advective and diffusive components. [ABSTRACT FROM AUTHOR]

Highlights • Developed a robust shock capturing method for DG schemes. • Intraelement variations are used for shock detection. • Smooth artificial viscosity is used for shock stabilization. • Benchmarked heating predictions against FV solvers in hypersonic viscous flows. • DG predictions exhibit reduced sensitivity to mesh topology and flux functions. Abstract This study is concerned with predicting surface heat transfer in viscous hypersonic flows using high-order discontinuous Galerkin (DG) methods. Currently, finite-volume (FV) schemes are most commonly employed for computing flows in which surface heat transfer is a target quantity; however, these schemes suffer from large sensitivities to a variety of factors, such as the inviscid flux function and the computational mesh. High-order DG methods offer advantages that can mitigate these sensitivities. As such, a simple and robust shock capturing method is developed for DG schemes. The method combines intraelement variations for shock detection with smooth artificial viscosity (AV) for shock stabilization. A parametric study is performed to evaluate the effects of AV on the solution. The shock capturing method is employed to accurately compute double Mach reflection and viscous hypersonic flows over a circular half-cylinder and a double cone, the latter of which involves a complex flow topology with multiple shock interactions and flow separation. Results show this methodology to be significantly less sensitive than FV schemes to mesh topology and inviscid flux function. Furthermore, quantitative comparisons with state-of-the-art FV calculations from an error vs. cost perspective are provided. [ABSTRACT FROM AUTHOR]

An original 2nd-order Method of Characteristics (MOC) which it works with a five-point interpolating scheme valid for solving the hyperbolic and quasilinear partial differential equations that describe the transient flow phenomenon in pipelines is shown. The results obtained by both MOC 2nd-order and exact solution (MOC 1st-order) are compared. It is shown that MOC 2nd-order allows obtain near-to-exact results within a wide range of Courant numbers. [ABSTRACT FROM AUTHOR]

Abstract We analyze the properties of a number of forms of artificial viscosity, when applied to the Lagrangian phase of a staggered grid Lagrange–remap hydrodynamics code, using a full von Neumann stability analysis. This allows us to derive the numerical dispersion relation for a single step of the scheme. We study the development of shocks within a weakly nonlinear acoustic wave in detail. The second-order von Neumann–Richtmyer artificial viscosity leads to significant post-shock ringing, while adding linear artificial viscosity reduces the accuracy of the scheme to first order and introduces a dependence on flow Mach number. The form of the numerical dispersion relation suggests that to control the post-shock ringing without reducing the accuracy of the solution, a higher-order artificial viscosity is required which will preferentially damp modes close to the mesh scale which are subject to significant phase error. The application of ideas based on the Large Eddy Simulation technique for the modelling of turbulent flows leads to a form of artificial viscous stress similar to that described by Schulz (1964). This artificial stress has several features which are beneficial for accurate modelling of compressible turbulent mixing processes, as demonstrated by a range of other test problems. We show that, in the limit of smooth flows, Christensen's approach of applying limiters to the components of a low-order artificial viscosity is analytically equivalent to a high-order scheme. However, the directionally-split implementation suggested in the original publication leads to damping of low Mach number Kelvin–Helmholtz roll up problems similar to that previously observed for finite-volume Godunov schemes. Highlights • Ringing in a finite difference scheme is explained using its dispersion relation. • Shock formation for Burgers equation is related to the turbulent inertial range. • An artificial viscosity is found by analogy to Large Eddy Simulation of turbulence. [ABSTRACT FROM AUTHOR]

Physics-informed Neural Network (PINN) is a promising tool that has been applied in a variety of physical phenomena described by partial differential equations (PDE). However, it has been observed that PINNs are difficult to train in certain "stiff" problems, which include various nonlinear hyperbolic PDEs that display shocks in their solutions. Recent studies added a diffusion term to the PDE, and an artificial viscosity (AV) value was manually tuned to allow PINNs to solve these problems. In this paper, we propose three approaches to address this problem, none of which rely on an a priori definition of the artificial viscosity value. The first method learns a global AV value, whereas the other two learn localized AV values around the shocks, by means of a parametrized AV map or a residual-based AV map. We applied the proposed methods to the inviscid Burgers equation and the Buckley-Leverett equation, the latter being a classical problem in Petroleum Engineering. The results show that the proposed methods are able to learn both a small AV value and the accurate shock location and improve the approximation error over a nonadaptive global AV alternative method. • Methods to add adaptive artificial viscosity to train PINNs to solve hyperbolic PDEs with shocks. • Adaptive methods accurately learn both the value and location of the artificial viscosity. • Successfully solved the Inviscid Burgers and Buckley-Leverett equations. • Additional computation cost over the baseline PINN is small. [ABSTRACT FROM AUTHOR]

The use of an artificial bulk viscosity for shock stabilization is a common approach employed in turbulence simulations with high-order numerics. The effect of the artificial bulk viscosity is analyzed in the context of large eddy simulations by using as a test case simulations of linearly-forced compressible homogeneous turbulence (Petersen and Livescu, 2010 [12] ). This case is unique in that it allows for the specification of a priori target values for total dissipation and ratio of solenoidal to dilatational dissipation. A comparison between these target values and the true predicted levels of dissipation is thus used to investigate the performance of the artificial bulk viscosity. Results show that the artificial bulk viscosity is effective at achieving stable solutions, but also leads to large values of artificial dissipation that outweigh the physical dissipation caused by fluid viscosity. An alternate approach, which employs the artificial thermal conductivity only, shows that the dissipation of dilatational modes is entirely due to the fluid viscosity. However, this method leads to unwanted Gibbs oscillations around the shocklets. The use of shock sensors that further localize the artificial bulk viscosity did not reduce the amount of artificial dissipation introduced by the artificial bulk viscosity. Finally, an improved forcing function that explicitly accounts for the role of the artificial bulk viscosity in the budget of turbulent kinetic energy was explored. [ABSTRACT FROM AUTHOR]

This work aims at developing a high-order numerical method for the propagation of acoustic shock waves using the discontinuous Galerkin method. High order methods tend to amplify the formation of spurious oscillations (Gibbs phenomenon) around the discontinuities/shocks, associated to the relative importance of higher-harmonics resulting from nonlinear propagation (in our case). To handle this critical issue, a new shock sensor is introduced for the sub-cell shock capturing. Thereafter, an element-centered smooth artificial viscosity is introduced into the system wherever an acoustic shock wave is sensed. Validation tests in 1D and 2D configurations show that the method is well-suited for the propagation of acoustic shock waves along with other physical effects like geometrical spreading and diffraction. [ABSTRACT FROM AUTHOR]

A lattice Boltzmann model on Hermite basis for compressible viscous flows is presented in this paper. The model is developed in the framework of double-distribution-function approach, which has adjustable specific-heat ratio and Prandtl number. It contains a density distribution function for the flow field and a total energy distribution function for the temperature field. The equilibrium distribution function is determined by Hermite expansion, and the D3Q27 and D3Q39 three-dimensional (3D) discrete velocity models are used, in which the discrete velocity model can be replaced easily. Moreover, an artificial viscosity is introduced to enhance the model for capturing shock waves. The model is tested through several cases of compressible flows, including 3D supersonic viscous flows with boundary layer. The effect of artificial viscosity is estimated. Besides, D3Q27 and D3Q39 models are further compared in the present platform. [ABSTRACT FROM AUTHOR]

The idea of using velocity dilation for shock capturing is revisited in this paper, combined with the discontinuous Galerkin method. The value of artificial viscosity is determined using direct dilation instead of its higher order derivatives to reduce cost and degree of difficulty in computing derivatives. Alternative methods for estimating the element size of large aspect ratio and smooth artificial viscosity are proposed to further improve robustness and accuracy of the model. Several benchmark tests are conducted, ranging from subsonic to hypersonic flows involving strong shocks. Instead of adjusting empirical parameters to achieve optimum results for each case, all tests use a constant parameter for the model with reasonable success, indicating excellent robustness of the method. The model is only limited to third-order accuracy for smooth flows. This limitation may be relaxed by using a switch or a wall function. Overall, the model is a good candidate for compressible flows with potentials of further improvement. [ABSTRACT FROM AUTHOR]

This paper presents the development of a cylindrical SPH formulation based on previous study of the literature (Petschek et al) with an explicit formulation for the artificial viscosity. The entire development is explained to propose a formulation adapted to solve Euler equations in the case of a Riemann problem with axis-symmetric conditions. Thus, the artificial viscosity is constructed to find smooth solutions of well-known Riemann problems such as Sod, Noh and Sedov problems. Numerical results are compared to exact solutions and observations are made on numerical parameters influence. This study contributes to validate the axis-symmetrical formulation for pure hydrodynamics tests. [ABSTRACT FROM AUTHOR]

Smoothed particle hydrodynamics (SPH) is a meshfree, Lagrangian particle method which has advantages in handling solids with extremely large deformation. Like any other numerical methods, cares must be taken to ensure its desirable accuracy and stability through considering several correction techniques in calculation. The selection of values for parameters in those correction approaches is a key step in SPH simulation, which is always difficult for new beginners to deal well with effectively. This paper examines the common inconsistency and instability problems in SPH method and studies its computational efficiency when applied to hydrodynamics problems with material strength like soil column collapse. We analyzed in detail how the correction techniques mitigate these inconsistency and instability problems. Also, the numerical testing results associate with different values for the parameters used in the correction techniques are provided for better understanding the influence of these parameters and for finding out the desirable values. It is found that (1) the SPH method is easily subjected to an inconsistency problem in the boundary area due to the boundary deficiency, and it can be treated well by adopting “virtual particles” contributing to the particle summations. (2) The numerical oscillation in SPH simulation can be mitigated effectively by artificial viscosity with the suggested parameter values. (3) The tension cracking treatment, artificial viscosity and artificial stress work well in removing the tensile instability problem in SPH method. In addition, the nearest neighboring particle searching (NNPS) algorithm, spacing ratio, smoothing length and time step influence the efficiency and accuracy of SPH method significantly. It is shown that SPH method with suggested parameters values can produce a very good result compared with the experimental result. [ABSTRACT FROM AUTHOR]

This work presents a new approach for curing the carbuncle instability. The idea underlying the approach is to introduce some dissipation in the form of right-hand sides of the Navier–Stokes equations into the basic method of solving Euler equations; in so doing, we replace the molecular viscosity coefficient by the artificial viscosity coefficient and calculate heat conductivity assuming that the Prandtl number is constant. For the artificial viscosity coefficient we have chosen a formula that is consistent with the von Neumann and Richtmyer artificial viscosity, but has its specific features (extension to multidimensional simulations, introduction of a threshold compression intensity that restricts additional dissipation to the shock layer only). The coefficients and the expression for the characteristic mesh size in this formula are chosen from a large number of Quirk-type problem computations. The new cure for the carbuncle flaw has been tested on first-order schemes (Godunov, Roe, HLLC and AUSM + schemes) as applied to one- and two-dimensional simulations on smooth structured grids. Its efficiency has been demonstrated on several well-known test problems. [ABSTRACT FROM AUTHOR]

A method is presented, that combines the defect and deferred correction approaches to approximate solutions of Navier-Stokes equations at high Reynolds number. The method is of high accuracy in both space and time, and it allows for the usage of legacy codes a frequent requirement in the simulation of turbulent flows in complex geometries. The two-step method is considered here; to obtain a regularization that is second order accurate in space and time, the method computes a low-order accurate, stable, and computationally inexpensive approximation (Backward Euler with artificial viscosity) twice. The results are readily extendable to the higher order accuracy cases by adding more correction steps. Both the theoretical results and the numerical tests provided demonstrate that the computed solution is stable and the accuracy in both space and time is improved after the correction step. We also perform a qualitative test to demonstrate that the method is capable of capturing qualitative features of a turbulent flow, even on a very coarse mesh. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 814-839, 2017 [ABSTRACT FROM AUTHOR]

A spatio-temporal adaptive artificial viscosity (AV) based shock-capturing scheme is proposed for the solution of both inviscid and viscous compressible flows using a high-order parallel Discontinuous Spectral Element Method (DSEM). The artificial viscosity and artificial thermal conduction coefficients are proportional to the viscous and thermal entropy generating terms, respectively, in the viscous entropy conservation law. The magnitude of AV is limited based on the explicit stable CFL criterion, so that the stable artificial viscous time step size is greater than the convective stable time step size. To further ensure the stability of this explicit approach, an adaptive variable order exponential filter is applied, if necessary, in elements where the AV has been limited. In viscous flow computations a modified Jameson's sensor (Ducros et al., 1999 [61] ) limits the AV to small values in viscous shear regions, so as to maintain a high-order resolution in smooth regions and an essentially non-oscillatory behavior near sharp gradients/shocks regions. We have performed a systematic and extensive validation of the algorithm with one-dimensional problems (inviscid moving shock and viscous shock-structure interaction), two-dimensional problems (inviscid steady and unsteady shocked flows and viscous shock-boundary layer interaction), and a three-dimensional supersonic turbulent flow over a ramped cavity. These examples demonstrate that the explicit DSEM scheme with adaptive artificial viscosity terms is stable, accurate and efficient. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a fourth-order accurate finite element method for incompressible variable density flow. The method is implicit in time and constructed with the Taylor series technique, and uses standard high-order Lagrange basis functions in space. Taylor series time-stepping relies on time derivative correction terms to achieve high-order accuracy. We provide detailed algorithms to approximate the time derivatives of the variable density Navier–Stokes equations. Numerical validations confirm a fourth-order accuracy for smooth problems. We also numerically illustrate that the Taylor series method is unsuitable for problems where regularity is lost by solving the 2D Rayleigh–Taylor instability problem. [ABSTRACT FROM AUTHOR]