1,721,021 research outputs found
An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics
No full textIn this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high-order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space-time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element-local space-time Galerkin finite element predictor on moving curved meshes to obtain a high-order accurate one-step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tnis connected to the new one at tn + 1by straight edges, hence providing unstructured space-time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature-free integration, in which the space-time boundaries of the control volumes are split into simplex sub-elements that yield constant space-time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub-elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high-order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd
High Order Direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume Schemes for Hyperbolic Systems on Unstructured Meshes
No full textIn this work we develop a new class of high order accurate Arbitrary-Lagrangian-Eulerian (ALE) one-step finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering moving unstructured triangular and tetrahedral meshes, respectively. As usual for finite volume schemes, data are represented within each control volume by piecewise constant values that evolve in time, hence implying the use of some strategies to improve the order of accuracy of the algorithm. In our approach high order of accuracy in space is obtained by adopting a WENO reconstruction technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space-time finite element predictor, which performs a one-step time discretization. Specifically, we adopt either the continuous Galerkin (CG) predictor, which does not allow discontinuities in time and is suitable for smooth time evolutions, or the discontinuous Galerkin (DG) predictor which can handle stiff source terms that might produce jumps in the local space-time solution. Since we are dealing with moving meshes the elements deform while the solution is evolving in time, hence making the use of a reference system very convenient. Therefore, within the space-time predictor, the physical element is mapped onto a reference element using a high order isoparametric approach, where the space-time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space-time nodes. The computational mesh continuously changes its configuration in time, following as closely as possible the flow motion. The entire mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. In order to obtain a continuous mesh configuration at any time level, the mesh motion is evaluated by assigning each node of the computational mesh with a unique velocity vector at each timestep. The node solver algorithm preforms the Lagrangian stage, while we rely on a rezoning algorithm to improve the mesh quality when the flow motion becomes very complex, hence producing highly deformed computational elements. A so-called relaxation algorithm is finally employed to partially recover the optimal Lagrangian accuracy where the computational elements are not distorted too much. We underline that our scheme is supposed to be an ALE algorithm, where the local mesh velocity can be chosen independently from the local fluid velocity. Once the vertex velocity and thus the new node location has been determined, the old element configuration is connected with the new one at the future time level with straight edges to represent the local mesh motion, in order to maintain algorithmic simplicity. The final ALE finite volume scheme is based directly on a space-time conservation formulation of the governing system of hyperbolic balance laws. The nonlinear system is reformulated more compactly using a space-time divergence operator and is then integrated on a moving space-time control volume. We adopt a linear parametrization of the space-time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. In our algorithm either a simple and robust Rusanov-type numerical flux or a more sophisticated and less dissipative Osher-type numerical flux is employed.
We apply the new high order direct ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical and relativistic magneto-hydrodynamics (MHD) equations and the non-conservative seven-equation Baer-Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes.
Furthermore we focus on the following issues to improve the algorithm efficiency: the time evolution, the numerical flux computation across element boundaries and the high order WENO reconstruction procedure. First, a time-accurate local time stepping (LTS) algorithm for unstructured triangular meshes is derived and presented, where each element can run at its own optimal time step, given by a local CFL stability condition. Then, we propose a new and efficient quadrature-free formulation for the flux computation, in which the space-time boundaries of each element are split into simplex sub-elements. This leads to space-time normal vectors as well as Jacobian matrices that are constant within each sub-element, hence allowing the flux integrals to be evaluated on the space-time reference control volume once and for all analytically during a preprocessing step. Finally, we consider the very new a posteriori MOOD paradigm, recently proposed for the Eulerian framework, to overcome the expensive WENO approach on moving meshes. The MOOD technique requires the use of only one central reconstruction stencil because the limiting procedure is carried out a posteriori instead of a priori, as done in the WENO formulation
A direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D
No full textIn this paper we present a new family of high order accurate Arbitrary-Lagrangian-Eulerian (ALE) one-step ADER-WENO finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations with stiff source terms on moving tetrahedral meshes in three space dimensions. A WENO reconstruction technique is used to achieve high order of accuracy in space, while an element-local space-time Discontinuous Galerkin finite element predictor on moving curved meshes is used to obtain a high order accurate one-step time discretization. Within the space-time predictor the physical element is mapped onto a reference element using a high order isoparametric approach, where the space-time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space-time nodes. Since our algorithm is cell-centered, the final mesh motion is computed by using a suitable node solver algorithm. A rezoning step as well as a flattener strategy are used in some of the test problems to avoid mesh tangling or excessive element deformations that may occur when the computation involves strong shocks or shear waves. The ALE algorithm presented in this article belongs to the so-called direct ALE methods because the final Lagrangian finite volume scheme is based directly on a space-time conservation formulation of the governing PDE system, with the rezoned geometry taken already into account during the computation of the fluxes.We apply our new high order unstructured ALE schemes to the 3D Euler equations of compressible gas dynamics, for which a set of classical numerical test problems has been solved and for which convergence rates up to sixth order of accuracy in space and time have been obtained. We furthermore consider the equations of classical ideal magnetohydrodynamics (MHD) as well as the non-conservative seven-equation Baer-Nunziato model of compressible multi-phase flows with stiff relaxation source terms. © 2014 Elsevier Inc
High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: Applications to compressible multi-phase flows
No full textIn this article we present the first better than second order accurate unstructured Lagrangian-type one-step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non-conservative products. The method achieves high order of accuracy in space together with essentially non-oscillatory behavior using a non-linear WENO reconstruction operator on unstructured triangular meshes. High order accuracy in time is obtained via a local Lagrangian space-time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one-step finite volume scheme is derived by integration over a moving space-time control volume, where the non-conservative products are treated by a path-conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian (ALE) method, where the mesh velocity can be chosen independently of the fluid velocity. The new scheme is applied to the full seven-equation Baer-Nunziato model of compressible multi-phase flows with relaxation source terms in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer-Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided. © 2013 Elsevier Ltd
High Order Accurate Direct Arbitrary-Lagrangian-Eulerian ADER-MOOD Finite Volume Schemes for Non-Conservative Hyperbolic Systems with Stiff Source Terms
No full textIn this paper we present a 2D/3D high order accurate finite volume scheme in the context of direct Arbitrary-Lagrangian-Eulerian algorithms for general hyperbolic systems of partial differential equations with non-conservative products and stiff source terms. This scheme is constructed with a single stencil polynomial reconstruction operator, a one-step space-time ADER integration which is suitably designed for dealing even with stiff sources, a nodal solver with relaxation to determine the mesh motion, a path-conservative integration technique for the treatment of non-conservative products and an a posteriori stabilization procedure derived from the so-called Multidimensional Optimal Order Detection (MOOD) paradigm. In this work we consider the seven equation Baer-Nunziato model of compressible multi-phase flows as a representative model involving non-conservative products as well as relaxation source terms which are allowed to become stiff. The new scheme is validated against a set of test cases on 2D/3D unstructured moving meshes on parallel machines and the high order of accuracy achieved by the method is demonstrated by performing a numerical convergence study. Classical Riemann problems and explosion problems with exact solutions are simulated in 2D and 3D. The overall numerical code is also profiled to provide an estimate of the computational cost required by each component of the whole algorithm
Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes
No full textIn this article we present a new class of high order accurate Arbitrary- Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in [25]. For that purpose, a new element-local weak formulation of the governing PDE is adopted on moving space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges. This is possible in the ALE framework, which allows a local mesh velocity that is different from the local fluid velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics. © 2013 Global-Science Press
High order accurate direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on moving curvilinear unstructured meshes
No full textIn this article we present a new high order accurate fully discrete one-step Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on moving unstructured curvilinear meshes in two and three space dimensions. The WENO reconstruction technique that is used to achieve high order of accuracy in space is performed on curved isoparametric triangular and tetrahedral elements, which are not necessarily defined by straight boundaries. High order of accuracy in time is obtained via an element-local space-time Galerkin finite element predictor on moving curved meshes already developed in [Boscheri W, Dumbser M. A direct arbitrary-lagrangian-eulerian ader-weno finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3d. Journal of Computational Physics 2014;275(0):484–523.]. Our algorithm belongs to the category of cell-centered schemes, therefore a nodal solver is used to compute the velocity at each vertex of the computational grid, as well as at each additional degree of freedom that is needed to approximate the curvilinear geometry. To avoid mesh tangling or extremely distorted elements, we propose to use a modified version of the rezoning algorithm presented in [Galera S, Maire P, Breil J. A two-dimensional unstructured cell-centered multi-material ale scheme using vof interface reconstruction. Journal of Computational Physics 2010;229:5755-5787.], which can deal with curvilinear elements in multiple space dimensions. The rezoned geometry is then taken into account directly during the computation of the fluxes, thus the resulting finite volume scheme is a direct ALE method based on a space-time conservation formulation of the governing PDE system. The space-time control volume is defined for each element at each time step adopting an isoparametric approach, i.e. relying on a set of space-time basis functions which are as accurate as the desired order of the scheme. In this way the numerical solution and the geometry configuration of each element are approximated with the same accuracy in space and time. The resulting scheme is thus high order accurate and fully-discrete in one single step, which is typical for the ADER approach. We apply our new algorithm to the Euler equations of compressible gas dynamics in two and three space dimensions, considering a set of classical numerical test problems on moving meshes. Furthermore numerical convergence studies show the high order of accuracy of the proposed method up to fifth order in space and time
Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers
No full textIn this paper we use the genuinely multidimensional HLL Riemann solvers recently developed by Balsara et al. in [13] to construct a new class of computationally efficient high order Lagrangian ADER-WENO one-step ALE finite volume schemes on unstructured triangular meshes. A nonlinear WENO reconstruction operator allows the algorithm to achieve high order of accuracy in space, while high order of accuracy in time is obtained by the use of an ADER time-stepping technique based on a local space-time Galerkin predictor. The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the grid, considering the entire Voronoi neighborhood of each node and allow for larger time steps than conventional one-dimensional Riemann solvers. The results produced by the multidimensional Riemann solver are then used twice in our one-step ALE algorithm: first, as a node solver that assigns a unique velocity vector to each vertex, in order to preserve the continuity of the computational mesh; second, as a building block for genuinely multidimensional numerical flux evaluation that allows the scheme to run with larger time steps compared to conventional finite volume schemes that use classical one-dimensional Riemann solvers in normal direction. The space-time flux integral computation is carried out at the boundaries of each triangular space-time control volume using the Simpson quadrature rule in space and Gauss-Legendre quadrature in time. A rezoning step may be necessary in order to overcome element overlapping or crossing-over. Since our one-step ALE finite volume scheme is based directly on a space-time conservation formulation of the governing PDE system, the remapping stage is not needed, making our algorithm a so-called direct ALE method.We apply the method presented in this article to two systems of hyperbolic conservation laws, namely the Euler equations of compressible gas dynamics and the equations of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to fourth order of accuracy in space and time have been carried out. Several numerical test problems have been solved to validate the new approach. Furthermore, the new high order Lagrangian schemes based on genuinely multidimensional Riemann solvers have been carefully compared with high order Lagrangian finite volume schemes based on conventional one-dimensional Riemann solvers. It has been clearly shown that due to the less restrictive CFL condition the new schemes based on multidimensional HLL and HLLC Riemann solvers are computationally more efficient than the ones based on a conventional one-dimensional Riemann solver technique. © 2014 Elsevier Inc
High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes
No full textWe present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on unstructured triangular meshes that is high order accurate in space and time and that also allows for time-accurate local time stepping (LTS). It extends our previous investigations on high order Lagrangian finite volume schemes with LTS carried out in [46] in one space dimension. The new scheme uses the following basic ingredients: a high order WENO reconstruction in space on unstructured meshes, an element-local high-order accurate space-time Galerkin predictor that performs the time evolution of the reconstructed polynomials within each element, the computation of numerical ALE fluxes at the moving element interfaces through approximate Riemann solvers, and a one-step finite volume scheme for the time update which is directly based on the integral form of the conservation equations in space-time. The inclusion of the LTS algorithm requires a number of crucial extensions, such as a proper scheduling criterion for the time update of each element and for each node; a virtual projection of the elements contained in the reconstruction stencils of the element that has to perform the WENO reconstruction; and the proper computation of the fluxes through the space-time boundary surfaces that will inevitably contain hanging nodes in time due to the LTS algorithm. We have validated our new unstructured Lagrangian LTS approach over a wide sample of test cases solving the Euler equations of compressible gas dynamics in two space dimensions, including shock tube problems, cylindrical explosion problems, as well as specific tests typically adopted in Lagrangian calculations, such as the Kidder, the Saltzman and the Sedov problem. When compared to the traditional global time stepping (GTS) method, the newly proposed LTS algorithm allows to reduce the number of element updates in a given simulation by a factor that may depend on the complexity of the dynamics, but which can be as large as ~4.7. Finally, we have also shown the improvement in terms of computational efficiency in a representative test for the special relativistic magnetohydrodynamics (RMHD) equations
Arbitrary-Lagrangian–Eulerian Discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes
No full textWe present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes, like molecular viscosity or heat conduction. High order piecewise polynomials of degree N are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach, making use of an element-local space–time Galerkin finite element predictor. A novel nodal solver algorithm based on the HLL flux is derived to compute the velocity for each nodal degree of freedom that describes the current mesh geometry. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Each technique generates a corresponding number of geometrical degrees of freedom needed to describe the current mesh configuration and which must be considered by the nodal solver for determining the grid velocity. The connection of the old mesh configuration at time tn with the new one at time tn+1 provides the space–time control volumes on which the governing equations have to be integrated in order to obtain the time evolution of the discrete solution. Our numerical method belongs to the category of so-called direct Arbitrary-Lagrangian–Eulerian (ALE) schemes, where a space–time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry (including a possible rezoning step) directly during the computation of the numerical fluxes. We emphasize that our method is a moving mesh method, as opposed to total Lagrangian formulations that are based on a fixed computational grid and which instead evolve the mapping of the reference configuration to the current one. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method recently developed in [62] for fixed unstructured grids. In this approach, the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria, such as the positivity of pressure and density, the absence of floating point errors (NaN) and the satisfaction of a relaxed discrete maximum principle (DMP) in the sense of polynomials. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed at the aid of a more robust second order TVD finite volume scheme. To preserve the subcell resolution capability of the original DG scheme, the FV limiter is run on a sub-grid that is 2N+1 times finer compared to the mesh of the original unlimited DG scheme. The new subcell averages are then gathered back into a high order DG polynomial by a usual conservative finite volume reconstruction operator. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated in order to check the accuracy and the robustness of the proposed numerical method in the context of the Euler and Navier–Stokes equations for compressible gas dynamics, considering both inviscid and viscous fluids. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD)
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