A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates (original) (raw)
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We perform a comparison of mass conservation properties of the continuous (CG) and discontinuous (DG) Galerkin methods on non-conforming, dynamically adaptive meshes for two atmospheric test cases. The two methods are implemented in a unified way which allows for a direct comparison of the non-conforming edge treatment. We outline the implementation details of the non-conforming direct stiffness summation algorithm for the CG method and show that the mass conservation error is similar to the DG method. Both methods conserve to machine precision, regardless of the presence of the non-conforming edges. For lower order polynomials the CG method requires additional stabilization to run for very long simulation times. We addressed this issue by using filters and/or additional artificial viscosity.
A high-order finite-volume method for conservation laws on locally refined grids
Communications in Applied Mathematics and Computational Science, 2011
We present a fourth-order accurate finite-volume method for solving time-dependent hyperbolic systems of conservation laws on Cartesian grids with multiple levels of refinement. The underlying method is a generalization of that developed by Colella, Dorr, Hittinger and Martin (2009) to nonlinear systems, and is based on using fourth-order accurate quadratures for computing fluxes on faces, combined with fourth-order accurate Runge-Kutta discretization in time. To interpolate boundary conditions at refinement boundaries, we interpolate in time in a manner consistent with the individual stages of the Runge-Kutta method, and interpolate in space by solving a least-squares problem over a neighborhood of each target cell for the coefficients of a cubic polynomial. The method also uses a variation on the extremum-preserving limiter of Colella and Sekora (2008), as well as slope flattening and a fourth-order accurate artificial viscosity for strong shocks. We show that the resulting method is fourth-order accurate for smooth solutions, and is robust in the presence of complex combinations of shocks and smooth flows. 1. High-order finite-volume methods In the finite-volume approach, the spatial domain in ޒ D is discretized as a union of rectangular control volumes that covers the spatial domain. For Cartesian-grid finite-volume methods, a control volume V i takes the form V i = [i h, (i + u)h] for i ∈ ޚ D , u = (1, 1,. .. , 1), where h is the grid spacing. A finite-volume discretization of a partial differential equation is based on averaging that equation over control volumes, applying the divergence theorem to replace volume integrals by integrals over the boundary of the control volume,
A scalable high-order discontinuous Galerkin method for global atmospheric modeling*
Parallel Computational Fluid Dynamics 2006, 2007
A conservative 3-D discontinuous Galerkin (DG) baroclinic model has been developed in the NCAR High-Order Method Modeling Environment (HOMME) to investigate global atmospheric flows. The computational domain is a cubed-sphere free from coordinate singularities. The DG discretization uses a high-order nodal basis set of orthogonal Lagrange-Legendre polynomials and fluxes of inter-element boundaries are approximated with Lax-Friedrichs numerical flux. The vertical discretization follows the 1-D vertical Lagrangian coordinates approach combined with the cell-integrated semi-Lagrangian method to preserve conservative remapping. Time integration follows the third-order SSP-RK scheme. To valid proposed 3-D DG model, the baroclinic instability test suite proposed by Jablonowski and Williamson is investigated. Parallel performance is evaluated on IBM Blue Gene/L and IBM POWER5 p575 supercomputers.
Journal of Computational Physics, 2006
High-order triangle-based discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. The finite element-type area integrals are evaluated using order 2N Gauss cubature rules. This leads to a full mass matrix which, unlike for continuous Galerkin (CG) methods such as the spectral element (SE) method presented in Giraldo and Warburton [A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. Comput. Phys. 207 (2005) 129-150], is small, local and efficient to invert. Two types of finite volume-type flux integrals are studied: a set based on Gauss-Lobatto quadrature points (order 2N À 1) and a set based on Gauss quadrature points (order 2N). Furthermore, we explore conservation and advection forms as well as strong and weak forms. Seven test cases are used to compare the different methods including some with scale contractions and shock waves. All three strong forms performed extremely well with the strong conservation form with 2N integration being the most accurate of the four DG methods studied. The strong advection form with 2N integration performed extremely well even for flows with shock waves. The strong conservation form with 2N À 1 integration yielded results almost as good as those with 2N while being less expensive. All the DG methods performed better than the SE method for almost all the test cases, especially for those with strong discontinuities. Finally, the DG methods required less computing time than the SE method due to the local nature of the mass matrix.
Study for the computational resolution of conservation equations of mass, momentum and energy
2019
48. double lambda_B=200; // Conductivitat térmica del material B, s uposant zinc 140 49. double lambda_C=200; 50. double lambda_D=200; 51. double sx=dist_y*1; 52. double sy=dist_x*1; 53. double Vol=sy*sx*1; // Volum d'un segment del material 54. double Vol_A=Vol; 55. double Vol_B=Vol; 56. double Vol_C=Vol; 57. double Vol_D=Vol; 58. double Tdiff; // Diferencia entre temperatura utilitzada i recent calculada 59. // Coeficients de discretització 60. double an [Nodes_x][Nodes_y]; 61. double as [Nodes_x][Nodes_y]; 62. double ae [Nodes_x][Nodes_y]; 63. double ap [Nodes_x][Nodes_y]; 64. double aw [Nodes_x][Nodes_y]; 65. double bp [Nodes_x][Nodes_y]; 66. double r [Nodes_x][Nodes_y]; 67. double rx[Nodes_x][Nodes_y]; 68. double ry[Nodes_x][Nodes_y]; 69. //lambdes 70. double lambda_node[Nodes_x][Nodes_y]; 71. for(i=1; i<=Nodes_x; i++) // Posem una temperatura inic ial a cadascun dels nodes 72. { 73. for(j=1; j<=Nodes_y; j++) 74. { 75. if(((dist_x*(i-1)+0.5*dist_x)<=delta_x_A) && dist_y*(j-1)+0.5*dist_y<=delta_y_A){ 76. lambda_node[i][j]=lambda_A; 77. }else if(((dist_x*(i-1)+0.5*dist_x)>(H-delta_x_B)) && dist_y*(j-1)+0.5*dist_y<=delta_y_B){ 78. lambda_node[i][j]=lambda_B; 79. }else if(((dist_x*(i-1)+0.5*dist_x)<=(H-delta_x_C)) && dist_y*(j-1)+0.5*dist_y>(H-delta_y_C)){ 80. lambda_node[i][j]=lambda_C; 81. }else{ 82. lambda_node[i][j]=lambda_D; 83. } 84. } 85. } 86. //Mapa inicial de temperatures (aribtrari) per a t=0; 87. double T1[Nodes_x][Nodes_y]; // Temperatura instant i 88. double T2[Nodes_x][Nodes_y]; // Temperatura instant i+1 89. double Tcalc[Nodes_x][Nodes_y]; // Auxiliar per adjudicar la T recent calculada a la T1 90. for(i=1; i<=Nodes_x; i++) // Posem una temperatura inic ial a cadascun dels nodes 91. { 92. for(j=1; j<=Nodes_y; j++) 93. { 94. T1[i][j]=273; 95. } 96. } 97. 98. // Amb les propietats definides i calculades anteriorment, podem procedir a re alitzar l'algoritme de resolució. 99. while (final==false)
International Journal for Numerical Methods in Engineering, 2009
This paper presents a comparison between two high order methods. The first one is a high-order Finite Volume (FV) discretization on unstructured grids that uses a meshfree method (Moving Least Squares (MLS)) in order to construct a piecewise polynomial reconstruction and evaluate the viscous fluxes. The second method is a discontinuous Galerkin (DG) scheme. Numerical examples of inviscid and viscous flows are presented and the solutions are compared. The accuracy of both methods, for the same grid resolution, is similar, although the finite volume scheme is consistently more accurate in the present tests. Furthermore, the DG scheme requires a larger number of degrees of freedom than the FV-MLS method.
We consider the conserv.ation properties of a staggered-grid Lagrange formulation of the hydrodynamics equations (SGH). Hydrodynamics algorithms are often formulated in a relatively ad hoc manner in which independent discretizations are proposedfor mass, momentum, energy, and so forth. We show that, once discretizations for mass and momentum are stated, the remaining discretizations are very nearly uniquely determined, so there is very liule latitude for variation. As has been known for some time, the kinetic energy discretization must follow directly from the momentum equation; and the internal energy must follow directly from the energy currents affecting the kinetic energy. A fundamental requirement (termed isentropicity)for numerical hydrodynamics algorithms is the ability to remain on an isentrope in the absence of heating or viscous forces and in the limit of small timesteps. We show that the requirements of energy conservation and isentropicity lead to the replacement of the usual volume calculation with a conservation integral. They further forbid the use of higher order functional representations for either velocity or stress within zones or control volumes, forcing the use of a constant stress element and a constant velocity control volume. This, in turn, causes the point and zone coordinates toformally disappear from the Cartesian formulation. The form of the work equations and the requirement for dissipation by viscous forces strongly limits the possible algebraic forms for artificial viscosity. The momentum equation and a center-of-mass definition lead directly to an angular momentum conservation law that is satisfied by the system. With a few straightforward substitutions, the Cartesian formulation can be converted to a multidimensional curvilinear one. The formulation in 2D axisymmetric geometry preserves rotational symmetry.