Improvement on spherical symmetry in two-dimensional cylindrical coordinates for a class of control volume Lagrangian schemes

Division of Applied Mathematics, Brown University, 02912, Providence, RI
Communications in Computational Physics (Impact Factor: 1.94). 01/2012; 11.


In [14], Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates. These schemes use a node-based discretization of the numerical fluxes. The control volume version has several distinguished properties, including the conservation of mass, momentum and total energy and compatibility with the geometric conservation law (GCL). However it also has a lim-itation in that it cannot preserve spherical symmetry for one-dimensional spherical flow. An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry, at the price of sacrificing conservation of momentum. In this paper, we apply the methodology proposed in our recent work [8] to the first order control volume scheme of Maire in [14] to obtain the spherical symmetry property. The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid, and meanwhile it maintains its original good properties such as conservation and GCL. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry, non-oscillation and robustness properties.

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    • "To our knowledge, the Lagrangian schemes developed in [10] [40] are the first better than second order non-oscillatory Lagrangian finite volume schemes published so far. Further work of Cheng and Shu contains the construction of symmetrypreserving Lagrangian schemes, see [11] [12]. A completely different class of fully Lagrangian methods can be found in meshless particle schemes such as the SPH approach [26–29, 45], which has become very popular to simulate fluid motion in complex deforming domains due to its algorithmic simplicity and high versatility and flexibility. "
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    ABSTRACT: In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws. High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkin method recently proposed in Dumbser, Enaux, and Toro (2008). In the Lagrangian framework considered here, the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element. For the space-time basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points. The moving space-time elements are mapped to a reference element using an isoparametric approach, i.e. the space-time mapping is defined by the same basis functions as the weak solution of the PDE. We show some computational examples in one space-dimension for non-stiff and for stiff balance laws, in particular for the Euler equations of compressible gas dynamics, for the resistive relativistic MHD equations, and for the relativistic radiation hydrodynamics equations. Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.
    Preview · Article · Jul 2012 · Communications in Computational Physics
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    • "We have performed all the tests shown in [13] by the scheme (2.18) and obtained the expected identical results as those in [13]. In this subsection, we will give several new tests which have not been shown in our previous papers to further verify the performance of the scheme and to validate the new implementation. "

    Preview · Article · Jan 2012 · Communications in Computational Physics
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    • "The time step ∆t n is controlled by both the CFL condition and the criterion on the variation of the volume, see [17] [10] for more details. "
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    ABSTRACT: Lagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion (ICF), due to their dis-tinguished advantage in capturing material interfaces automatically. In some of these ap-plications, multiple internal energy equations such as those for electron, ion and radiation are involved. In the past decades, several staggered-grid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However such schemes are typically not conservative for the total energy. Recently, signif-icant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form of the total energy, therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications. Such modifications, if not designed carefully, may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy. In this paper, we establish an equivalency relationship between the cell-centered discretiza-tions of the Euler equations in the forms of the total energy and of the internal energy. By a carefully designed modification in the implementation, the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equa-tions and meanwhile it does not lose its total energy conservation property. An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrody-namic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry preserving, accuracy and non-oscillatory performance.
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