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Magnetohydrodynamics (MHD) studies the behaviors of ionized gases, such as plasmas, in the presence of a magnetic field. MHD is used in many applications, such as geophysics, space physics, and nuclear fusion. Despite intensive research in recent decades, many physical and numerical aspects of MHD are not well understood. The challenges inherent in solving MHD stem from the obstacles encountered in ordinary hydrodynamics, such as those described by the compressible Euler/Navier-Stokes equations, along with the intricacies arising from electromagnetism. A characteristic of compressible flows is their tendency to develop shocks/discontinuities over time. This often leads to unphysical traits in numerical approximations if the capturing scheme is not constructed properly. By physical laws, the magnetic field is solenoidal. However, in practice, numerical schemes seldom ensure this property precisely, which may lead to instability and convergence to wrong solutions. In numerical simulation of many applications, positive physical quantities such as density and pressure can easily become negative. On the whole, preserving the physical relevance of the numerical solutions poses a significant challenge in MHD. This thesis presents several numerical schemes based on Galerkin approximations to solve MHD. The schemes rely on viscous regularization, a technique to remove mathematical singularities by adding a vanishing viscosity term to the MHD equations. At the continuous level, we propose several choices of viscous regularization and rigorously show that they are consistent with thermodynamics. Based on these choices, we construct numerical schemes of which robustness is confirmed through many challenging benchmarks. Finally, we propose a nonconventional algorithm that simultaneously preserves many desirable physical properties, including positivity of density and internal energy, conservation of total energy, minimum entropy principle, and zero magnetic divergence.

Magnetohydrodynamics (MHD) studies the behaviors of ionized gases, such as plasmas, in the presence of a magnetic field. MHD is used in many applications, such as geophysics, space physics, and nuclear fusion. Despite intensive research in recent decades, many physical and numerical aspects of MHD are not well understood. The challenges inherent in solving MHD stem from the obstacles encountered in ordinary hydrodynamics, such as those described by the compressible Euler/Navier-Stokes equations, along with the intricacies arising from electromagnetism. A characteristic of compressible flows is their tendency to develop shocks/discontinuities over time. This often leads to unphysical traits in numerical approximations if the capturing scheme is not constructed properly. By physical laws, the magnetic field is solenoidal. However, in practice, numerical schemes seldom ensure this property precisely, which may lead to instability and convergence to wrong solutions. In numerical simulation of many applications, positive physical quantities such as density and pressure can easily become negative. On the whole, preserving the physical relevance of the numerical solutions poses a significant challenge in MHD. This thesis presents several numerical schemes based on Galerkin approximations to solve MHD. The schemes rely on viscous regularization, a technique to remove mathematical singularities by adding a vanishing viscosity term to the MHD equations. At the continuous level, we propose several choices of viscous regularization and rigorously show that they are consistent with thermodynamics. Based on these choices, we construct numerical schemes of which robustness is confirmed through many challenging benchmarks. Finally, we propose a nonconventional algorithm that simultaneously preserves many desirable physical properties, including positivity of density and internal energy, conservation of total energy, minimum entropy principle, and zero magnetic divergence.

Flooding has always been and is still today a disastrous event with agricultural, infrastructural, economical and not least humanitarian ramifications. Understanding the behaviour of floods is crucial to be able to prevent or mitigate future catastrophes, a task which can be accomplished by modelling the water flow. In this thesis the finite element method is employed to solve the shallow water equations, which govern water flow in shallow environments such as rivers, lakes and dams, a methodology that has been widely used for flooding simulations. Alternative approaches to model floods are however also briefly discussed. Since the finite element method suffers from numerical instabilities when solving nonlinear conservation laws, the shallow water equations are stabilised by introducing a high-order nonlinear artificial viscosity, constructed using a multi-mesh strategy. The accuracy, robustness and well-balancedness of the solution are examined through a variety of benchmark tests. Finally, the equations are extended to include a friction term, after which the effectiveness of the method in a real-life scenario is verified by a prolonged simulation of the Malpasset dam break.

Magnetohydrodynamics (MHD) studies the behaviors of ionized gases, such as plasmas, in the presence of a magnetic field. MHD is used in many applications, such as geophysics, space physics, and nuclear fusion. Despite intensive research in recent decades, many physical and numerical aspects of MHD are not well understood. The challenges inherent in solving MHD stem from the obstacles encountered in ordinary hydrodynamics, such as those described by the compressible Euler/Navier-Stokes equations, along with the intricacies arising from electromagnetism. A characteristic of compressible flows is their tendency to develop shocks/discontinuities over time. This often leads to unphysical traits in numerical approximations if the capturing scheme is not constructed properly. By physical laws, the magnetic field is solenoidal. However, in practice, numerical schemes seldom ensure this property precisely, which may lead to instability and convergence to wrong solutions. In numerical simulation of many applications, positive physical quantities such as density and pressure can easily become negative. On the whole, preserving the physical relevance of the numerical solutions poses a significant challenge in MHD. This thesis presents several numerical schemes based on Galerkin approximations to solve MHD. The schemes rely on viscous regularization, a technique to remove mathematical singularities by adding a vanishing viscosity term to the MHD equations. At the continuous level, we propose several choices of viscous regularization and rigorously show that they are consistent with thermodynamics. Based on these choices, we construct numerical schemes of which robustness is confirmed through many challenging benchmarks. Finally, we propose a nonconventional algorithm that simultaneously preserves many desirable physical properties, including positivity of density and internal energy, conservation of total energy, minimum entropy principle, and zero magnetic divergence.

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.

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; 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.

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; 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.

Flooding has always been and is still today a disastrous event with agricultural, infrastructural, economical and not least humanitarian ramifications. Understanding the behaviour of floods is crucial to be able to prevent or mitigate future catastrophes, a task which can be accomplished by modelling the water flow. In this thesis the finite element method is employed to solve the shallow water equations, which govern water flow in shallow environments such as rivers, lakes and dams, a methodology that has been widely used for flooding simulations. Alternative approaches to model floods are however also briefly discussed. Since the finite element method suffers from numerical instabilities when solving nonlinear conservation laws, the shallow water equations are stabilised by introducing a high-order nonlinear artificial viscosity, constructed using a multi-mesh strategy. The accuracy, robustness and well-balancedness of the solution are examined through a variety of benchmark tests. Finally, the equations are extended to include a friction term, after which the effectiveness of the method in a real-life scenario is verified by a prolonged simulation of the Malpasset dam break.

The simulation of fluid flow is a challenging and important problem in science and engineering. This thesis primarily focuses on developing finite element methods for simulating subsonic two-phase flows with varying densities, described by the variable density incompressible Navier-Stokes equations. These equations are commonly used to model a wide range of phenomena, including aerodynamic forces around vehicles, climate and weather prediction, combustion and the spread of pollution. Incompressible flow is characterized by the velocity field satisfying the divergence-free condition. However, numerically satisfying this condition is one of the main challenges in simulating such flows. In practice, this condition is rarely satisfied exactly, which can result in stability and conservation issues in computations. Moreover, enforcing the divergence-free condition is a primary computational bottleneck for incompressible flow solvers. To improve computational efficiency, we explore and develop artificial compressibility techniques, which regularize this constraint. Additionally, we develop a new practical and useful formulation for variable density flow. This formulation allows Galerkin methods to enhance conservation properties when the divergence-free condition is not strongly enforced, leading to significantly improved accuracy and robustness. Another primary difficulty in simulating fluid flows arises from the challenge of accurately representing underresolved flows, where the mesh resolution cannot capture the gradient of the true solution. This leads to stability issues unless appropriate stabilization techniques are used. In this thesis, we develop new high-order accurate artificial viscosity techniques to deal with this issue. Furthermore, we thoroughly investigate the properties of viscous regularizations, ensuring that kinetic energy stability is guaranteed when using artificial viscosity.

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., eSSENCE - An eScience Collaboration

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., eSSENCE - An eScience Collaboration

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., eSSENCE - An eScience Collaboration

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., eSSENCE - An eScience Collaboration

The simulation of fluid flow is a challenging and important problem in science and engineering. This thesis primarily focuses on developing finite element methods for simulating subsonic two-phase flows with varying densities, described by the variable density incompressible Navier-Stokes equations. These equations are commonly used to model a wide range of phenomena, including aerodynamic forces around vehicles, climate and weather prediction, combustion and the spread of pollution. Incompressible flow is characterized by the velocity field satisfying the divergence-free condition. However, numerically satisfying this condition is one of the main challenges in simulating such flows. In practice, this condition is rarely satisfied exactly, which can result in stability and conservation issues in computations. Moreover, enforcing the divergence-free condition is a primary computational bottleneck for incompressible flow solvers. To improve computational efficiency, we explore and develop artificial compressibility techniques, which regularize this constraint. Additionally, we develop a new practical and useful formulation for variable density flow. This formulation allows Galerkin methods to enhance conservation properties when the divergence-free condition is not strongly enforced, leading to significantly improved accuracy and robustness. Another primary difficulty in simulating fluid flows arises from the challenge of accurately representing underresolved flows, where the mesh resolution cannot capture the gradient of the true solution. This leads to stability issues unless appropriate stabilization techniques are used. In this thesis, we develop new high-order accurate artificial viscosity techniques to deal with this issue. Furthermore, we thoroughly investigate the properties of viscous regularizations, ensuring that kinetic energy stability is guaranteed when using artificial viscosity.

The simulation of fluid flow is a challenging and important problem in science and engineering. This thesis primarily focuses on developing finite element methods for simulating subsonic two-phase flows with varying densities, described by the variable density incompressible Navier-Stokes equations. These equations are commonly used to model a wide range of phenomena, including aerodynamic forces around vehicles, climate and weather prediction, combustion and the spread of pollution. Incompressible flow is characterized by the velocity field satisfying the divergence-free condition. However, numerically satisfying this condition is one of the main challenges in simulating such flows. In practice, this condition is rarely satisfied exactly, which can result in stability and conservation issues in computations. Moreover, enforcing the divergence-free condition is a primary computational bottleneck for incompressible flow solvers. To improve computational efficiency, we explore and develop artificial compressibility techniques, which regularize this constraint. Additionally, we develop a new practical and useful formulation for variable density flow. This formulation allows Galerkin methods to enhance conservation properties when the divergence-free condition is not strongly enforced, leading to significantly improved accuracy and robustness. Another primary difficulty in simulating fluid flows arises from the challenge of accurately representing underresolved flows, where the mesh resolution cannot capture the gradient of the true solution. This leads to stability issues unless appropriate stabilization techniques are used. In this thesis, we develop new high-order accurate artificial viscosity techniques to deal with this issue. Furthermore, we thoroughly investigate the properties of viscous regularizations, ensuring that kinetic energy stability is guaranteed when using artificial viscosity.

The simulation of fluid flow is a challenging and important problem in science and engineering. This thesis primarily focuses on developing finite element methods for simulating subsonic two-phase flows with varying densities, described by the variable density incompressible Navier-Stokes equations. These equations are commonly used to model a wide range of phenomena, including aerodynamic forces around vehicles, climate and weather prediction, combustion and the spread of pollution. Incompressible flow is characterized by the velocity field satisfying the divergence-free condition. However, numerically satisfying this condition is one of the main challenges in simulating such flows. In practice, this condition is rarely satisfied exactly, which can result in stability and conservation issues in computations. Moreover, enforcing the divergence-free condition is a primary computational bottleneck for incompressible flow solvers. To improve computational efficiency, we explore and develop artificial compressibility techniques, which regularize this constraint. Additionally, we develop a new practical and useful formulation for variable density flow. This formulation allows Galerkin methods to enhance conservation properties when the divergence-free condition is not strongly enforced, leading to significantly improved accuracy and robustness. Another primary difficulty in simulating fluid flows arises from the challenge of accurately representing underresolved flows, where the mesh resolution cannot capture the gradient of the true solution. This leads to stability issues unless appropriate stabilization techniques are used. In this thesis, we develop new high-order accurate artificial viscosity techniques to deal with this issue. Furthermore, we thoroughly investigate the properties of viscous regularizations, ensuring that kinetic energy stability is guaranteed when using artificial viscosity.

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., eSSENCE - An eScience Collaboration

We show at the PDE level that the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. We then numerically investigate this regularization for the MHD equations using continuous finite elements in space and explicit strong stability preserving Runge–Kutta methods in time. The artificial viscosity coefficient of the regularization term is constructed to be proportional to the entropy residual of MHD. It is shown that the method has a high order of accuracy for smooth problems and captures strong shocks and discontinuities accurately for non-smooth problems.

We show at the PDE level that the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. We then numerically investigate this regularization for the MHD equations using continuous finite elements in space and explicit strong stability preserving Runge–Kutta methods in time. The artificial viscosity coefficient of the regularization term is constructed to be proportional to the entropy residual of MHD. It is shown that the method has a high order of accuracy for smooth problems and captures strong shocks and discontinuities accurately for non-smooth problems.

We show at the PDE level that the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. We then numerically investigate this regularization for the MHD equations using continuous finite elements in space and explicit strong stability preserving Runge–Kutta methods in time. The artificial viscosity coefficient of the regularization term is constructed to be proportional to the entropy residual of MHD. It is shown that the method has a high order of accuracy for smooth problems and captures strong shocks and discontinuities accurately for non-smooth problems.

We show at the PDE level that the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. We then numerically investigate this regularization for the MHD equations using continuous finite elements in space and explicit strong stability preserving Runge–Kutta methods in time. The artificial viscosity coefficient of the regularization term is constructed to be proportional to the entropy residual of MHD. It is shown that the method has a high order of accuracy for smooth problems and captures strong shocks and discontinuities accurately for non-smooth problems.

We show at the PDE level that the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. We then numerically investigate this regularization for the MHD equations using continuous finite elements in space and explicit strong stability preserving Runge–Kutta methods in time. The artificial viscosity coefficient of the regularization term is constructed to be proportional to the entropy residual of MHD. It is shown that the method has a high order of accuracy for smooth problems and captures strong shocks and discontinuities accurately for non-smooth problems.

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.

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 non-linear 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., This work has been supported by FEDER and the Spanish Government, Ministerio de Economía y Competitividad grant project contract CTM2014-55014-C3-3-R, Ministerio de Ciencia Innovación y Universidades grant project contract PGC2018-097257-B-C33 and the grant BES-2015-072833., Peer Reviewed, Postprint (author's final draft)

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.

The Viscous Garabedian and Korn (VGK) is a computational fluid dynamics (CFD) code, used for the prediction of viscous flows around two dimensional aerofoil sections (mapped onto a circular grid) in a subsonic free stream. The code is written in “Fortran 77” and solves the full potential flow equations using a finite difference algorithm. The code has been recently published commercially with a data preparation routine to facilitate its use [1]. The present work investigates the code’s consistency, stability and convergence in the hope that it will shed light on the code’s performance and limitations under various flow conditions and code’s numerical parameters.

The Viscous Garabedian and Korn (VGK) is a computational fluid dynamics (CFD) code, used for the prediction of viscous flows around two dimensional aerofoil sections (mapped onto a circular grid) in a subsonic free stream. The code is written in “Fortran 77” and solves the full potential flow equations using a finite difference algorithm. The code has been recently published commercially with a data preparation routine to facilitate its use [1]. The present work investigates the code’s consistency, stability and convergence in the hope that it will shed light on the code’s performance and limitations under various flow conditions and code’s numerical parameters.

The Viscous Garabedian and Korn (VGK) is a computational fluid dynamics (CFD) code, used for the prediction of viscous flows around two dimensional aerofoil sections (mapped onto a circular grid) in a subsonic free stream. The code is written in “Fortran 77” and solves the full potential flow equations using a finite difference algorithm. The code has been recently published commercially with a data preparation routine to facilitate its use [1]. The present work investigates the code’s consistency, stability and convergence in the hope that it will shed light on the code’s performance and limitations under various flow conditions and code’s numerical parameters.

The Viscous Garabedian and Korn (VGK) is a computational fluid dynamics (CFD) code, used for the prediction of viscous flows around two dimensional aerofoil sections (mapped onto a circular grid) in a subsonic free stream. The code is written in “Fortran 77” and solves the full potential flow equations using a finite difference algorithm. The code has been recently published commercially with a data preparation routine to facilitate its use [1]. The present work investigates the code’s consistency, stability and convergence in the hope that it will shed light on the code’s performance and limitations under various flow conditions and code’s numerical parameters.

The Viscous Garabedian and Korn (VGK) is a computational fluid dynamics (CFD) code, used for the prediction of viscous flows around two dimensional aerofoil sections (mapped onto a circular grid) in a subsonic free stream. The code is written in “Fortran 77” and solves the full potential flow equations using a finite difference algorithm. The code has been recently published commercially with a data preparation routine to facilitate its use [1]. The present work investigates the code’s consistency, stability and convergence in the hope that it will shed light on the code’s performance and limitations under various flow conditions and code’s numerical parameters.

Prediction of heat loads during hypersonic re-entry is of great interest in space exploration and in the topic of space debris as well. To date, there is no accurate method to reproduce either experimentally or numerically the physics in re-entry conditions. On the numerical side, high-order discontinuous Galerkin methods have potential to improve results achieved with state-of-the-art finite volume methods. However, they suffer from Gibbs-type oscillations around discontinuities such as shocks. This work uses artificial viscosity to tackle this issue. The artificial viscosity method used in this work contains three user-defined parameters which specify the magnitude, location and width of the artificial viscosity. A parametric study of these parameters on a steady inviscid wedge test case at Mach 2 showed that tuning their values is a non-trivial task and it appeared that in some cases shock smoothness and shock thickness need to be traded-off. Furthermore, it was observed that sufficient refinement in mesh size or polynomial order was needed to obtain satisfying results. The Sod test case was considered to test the method in an unsteady inviscid case. Good results were obtained, although the shock was better resolved than the contact discontinuity. The method was also tested on another unsteady inviscid test case, the Shu-Osher problem. The latter is more challenging due to the shock strength varying in time. Specifying the artificial viscosity parameters was challenging since the set had to account for variation of the shock strength in time. The test case also highlighted the need for a method that keeps variables within their physical bounds. Prediction of the heat flux on the surface of a half-cylinder in a hypersonic flow was attempted. However, this was unsuccessful and the Mach number was eventually reduced to 3.25. Strong non-physical oscillations appeared in the heat flux profile even with smooth contour plots and pressure profile at the surface o, Aerospace Engineering

Prediction of heat loads during hypersonic re-entry is of great interest in space exploration and in the topic of space debris as well. To date, there is no accurate method to reproduce either experimentally or numerically the physics in re-entry conditions. On the numerical side, high-order discontinuous Galerkin methods have potential to improve results achieved with state-of-the-art finite volume methods. However, they suffer from Gibbs-type oscillations around discontinuities such as shocks. This work uses artificial viscosity to tackle this issue. The artificial viscosity method used in this work contains three user-defined parameters which specify the magnitude, location and width of the artificial viscosity. A parametric study of these parameters on a steady inviscid wedge test case at Mach 2 showed that tuning their values is a non-trivial task and it appeared that in some cases shock smoothness and shock thickness need to be traded-off. Furthermore, it was observed that sufficient refinement in mesh size or polynomial order was needed to obtain satisfying results. The Sod test case was considered to test the method in an unsteady inviscid case. Good results were obtained, although the shock was better resolved than the contact discontinuity. The method was also tested on another unsteady inviscid test case, the Shu-Osher problem. The latter is more challenging due to the shock strength varying in time. Specifying the artificial viscosity parameters was challenging since the set had to account for variation of the shock strength in time. The test case also highlighted the need for a method that keeps variables within their physical bounds. Prediction of the heat flux on the surface of a half-cylinder in a hypersonic flow was attempted. However, this was unsuccessful and the Mach number was eventually reduced to 3.25. Strong non-physical oscillations appeared in the heat flux profile even with smooth contour plots and pressure profile at the surface o, Aerospace Engineering

In this paperwork, writing an Euler equation’s solver was described. The main purpose of this work is an investigation of mathematical models to solve inviscid and compressible fluid flow in two-dimensional tasks. During this research, I explored how to influence number of CFL and artificial viscosity on an accuracy and a stability of a solution., В данной работе было описано написание программного кода для решения уравнения Эйлера. Целью данной работы являлось получение навыков прикладного использования уравнения механики жидкости и газов, изучение математических моделей решений задач с истечением невязких сжимаемых жидкостей. Были определены влияния искусственной вязкости на соответствие физической картине истечения данных жидкостей. Также был проведен анализ влияния числа Куранта на сходимость расчета.

The capability of the smoothed-particle hydrodynamics (SPH) method to model supercritical flow in circular pipe bends is considered. The standard SPH method, which makes use of dynamic boundary particles (DBP), is supplemented with the original algorithm for the treatment of open boundaries. The method is assessed through a comparison with measured free-surface profiles in a pipe bend, and already proposed regression curves for estimation of the flow-type in a pipe bend. The sensitivity of the model to different parameters is also evaluated. It is shown that an adequate choice of the artificial viscosity coefficient and the initial particle spacing can lead to correct presentation of the flow-type in a bend. Due to easiness of its implementation, the SPH method can be efficiently used in the design of circular conduits with supercritical flow in a bend, such as tunnel spillways, and bottom outlets of dams, or storm sewers.

The capability of the smoothed-particle hydrodynamics (SPH) method to model supercritical flow in circular pipe bends is considered. The standard SPH method, which makes use of dynamic boundary particles (DBP), is supplemented with the original algorithm for the treatment of open boundaries. The method is assessed through a comparison with measured free-surface profiles in a pipe bend, and already proposed regression curves for estimation of the flow-type in a pipe bend. The sensitivity of the model to different parameters is also evaluated. It is shown that an adequate choice of the artificial viscosity coefficient and the initial particle spacing can lead to correct presentation of the flow-type in a bend. Due to easiness of its implementation, the SPH method can be efficiently used in the design of circular conduits with supercritical flow in a bend, such as tunnel spillways, and bottom outlets of dams, or storm sewers.

The capability of the smoothed-particle hydrodynamics (SPH) method to model supercritical flow in circular pipe bends is considered. The standard SPH method, which makes use of dynamic boundary particles (DBP), is supplemented with the original algorithm for the treatment of open boundaries. The method is assessed through a comparison with measured free-surface profiles in a pipe bend, and already proposed regression curves for estimation of the flow-type in a pipe bend. The sensitivity of the model to different parameters is also evaluated. It is shown that an adequate choice of the artificial viscosity coefficient and the initial particle spacing can lead to correct presentation of the flow-type in a bend. Due to easiness of its implementation, the SPH method can be efficiently used in the design of circular conduits with supercritical flow in a bend, such as tunnel spillways, and bottom outlets of dams, or storm sewers.

The capability of the smoothed-particle hydrodynamics (SPH) method to model supercritical flow in circular pipe bends is considered. The standard SPH method, which makes use of dynamic boundary particles (DBP), is supplemented with the original algorithm for the treatment of open boundaries. The method is assessed through a comparison with measured free-surface profiles in a pipe bend, and already proposed regression curves for estimation of the flow-type in a pipe bend. The sensitivity of the model to different parameters is also evaluated. It is shown that an adequate choice of the artificial viscosity coefficient and the initial particle spacing can lead to correct presentation of the flow-type in a bend. Due to easiness of its implementation, the SPH method can be efficiently used in the design of circular conduits with supercritical flow in a bend, such as tunnel spillways, and bottom outlets of dams, or storm sewers.

Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high-order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high-order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology., Peer Reviewed, Postprint (published version)

A shock capturing strategy for high order Discontinuous Galerkin methods for conservation laws is proposed. We present a method in the one-dimensional case based on the introduction of artificial viscosity into the original equations. With this approach the shock is capture with sharp resolution maintaining high-order accuracy. The ideas for the extension to the two-dimensional case are also set.

QC 20100903

Numerical computations show that solutions of hyperbolic conservation laws obtained by second or higher order shock capturing methods in many cases are only first order accurate downstream of shocks (see, e.g., [M. H. Carpenter and J.H. Casper, AIAA J., 37 (1999), pp. 1072 1079]). We use matched asymptotic expansions to analyze the degeneration in order of accuracy for stationary solutions of hyperbolic conservation laws in two space dimensions., QC 20100525

First order errors downstream of shocks have been detected in computations with higher order shock capturing schemes in one and two dimensions. Based on a matched asymptotic expansion analysis we show how to modify the artificial viscosity and raise the order of accuracy., QC 20100525

NR 20140805

The project was concerned with the development of high performance computer codes for problems in transonic aerodynamics. A practical rule to calculate the wave drag for solutions of the Euler equations was developed from an entropy equality. Symmetric shockless airfoils were analyzed for which uniqueness fails in the transonic case not just for potential flow, but also for the Euler equations.

This paper is a brief summary of an investigation using the finite element method in the study of wave propagation and impact problems in solid materials. Guidelines are provided for the determination of the computational grid and the shape and size of the elements. Two examples are presented to show effects of mesh refinements on solution accuracy. Problems concerning the effects of artificial viscosity, as well as pressure and strain changes in the orientation of the grid, are presented. This study was performed with EPIC-2, a computer code for elastic-plastic impact calculations in two dimensions. EPIC-2 makes use of triangular elements in several configurations. The results presented herein should serve as a guide to proper selection of mesh size and orientation for accurate representation of physical problems.

The hydrodynamics code HULL used by the Air Force Weapons Laboratory (AFWL) uses the Von Neumann-Richtmeyer method to maintain stability in the numerical computation of a propagating shock. This method adds a pseudoviscous term to the pressure in a compression region. Care is needed in the application of this method, however. Too little artificial viscosity will not maintain stability. Too much will 'smear' the shock over many zones, making the shock practically continuous. The AFWL has developed a relation which expresses the linear viscosity coefficient as a function of maximum density. The effect of this variable viscosity coefficient is to maintain stability while limiting the smear to 10 zones for overpressure levels of 2 psi to 1200 psi in air. (Author)

In this paper, we examine mathematical properties of an equation arising on the theory of ocean circulation. The existence of weak solutions to the equations proposed by Charney (1955) for the inertial model of the Gulf Stream are established by means of the method of artificial viscosity. Keywords: Navier-Stokes equations; Artificial viscosity.