In this work we present new second order semi-discrete central schemes for systems of hyperbolic conservation laws on curvilinear grids. Our methods generalise the two-dimensional central-upwind schemes developed by Kurganov and Tadmor. In these schemes we account for area and volume changes in the numerical flux functions due to the non-cartesian geometries. In case of vectorial conservation laws we introduce a general prescription of the geometrical source terms valid for various orthogonal curvilinear coordinate systems. The methods are applied to the two-dimensional Euler equations of inviscid gas dynamics with and without angular momentum transport. In the latter case we introduce a new test problem to examine the detailed conservation of specific angular momentum.
Applying high-order finite-difference schemes, like the extensively used linear-upwind or WENO schemes, to curvilinear grids can be problematic. The geometrically induced error from grid Jacobian and metrics evaluation can pollute the flow field, and degrade the accuracy or cause the simulation failure even when uniform flow imposed, i.e. free-stream preserving problem. In order to address this issue, a method for general linear-upwind and WENO schemes preserving free-stream on stationary curvilinear grids is proposed. Following Lax-Friedrichs splitting, this method rewrites the numerical flux…
In simulations of compressible flows, the conservative finite difference method (FDM) based on the nonlinear upwind schemes, e.g. WENO5, might violate free-stream preserving (FP), due to the loss of the geometric conservation law (GCL) identity when applied on the curvilinear grids. Although some techniques on FP have been proposed previously, no general rule is given for this issue. In this paper, by rearranging the upwind dissipation of the nonlinear schemes as a combination of sub-stencil reconstructions (taking WENO5 as an example), it can be proved that the upwind dissipation diminishes u…
In this paper, we develop a new free-stream preserving (FP) method for high-order upwind conservative finite-difference (FD) schemes on the curvilinear grids. This FP method is constrcuted by subtracting a reference cell-face flow state from each cell-center value in the local stencil of the original upwind conservative FD schemes, which effectively leads to a reformulated dissipation. It is convenient to implement this method, as it does not require to modify the original forms of the upwind schemes. In addition, the proposed method removes the constraint in the traditional FP conservative FD…
Based on the understandings regarding linear upwind schemes with flux splitting to achieve free-stream preservation (Q. Li, etc. Commun. Comput. Phys., 22 (2017) 64-94), a series of WENO interpolation-based and upwind-biased nonlinear schemes are proposed in this study. By means of engagement of fluxes on midpoints, the nonlinearity of schemes is introduced through WENO interpolations, and upwind-biased features are acquired through the choice of dependent grid stencil. Regarding the third- and fifth-order versions, schemes with one and two midpoints are devised and carefully tested. With the …
Cases have shown that WENO schemes usually behave robustly on problems containing shocks with high pressure ratios when uniformed or smooth grids are present, while nonlinear schemes based on WENO interpolations might relatively be liable to numerical instability. In the meanwhile, the latter have manifested their advantages in computations on grids of bad quality, because the free-stream preservation is easily realized there, and what is more flux-splitting schemes with low dissipations can be engaged inherently as well. Targeting at above dissatisfactions, a method by hybridizing WENO implem…
High order reconstruction in the finite volume (FV) approach is achieved by a more fundamental form of the fifth order WENO reconstruction in the framework of orthogonally-curvilinear coordinates, for solving the hyperbolic conservation equations. The derivation employs a piecewise parabolic polynomial approximation to the zone averaged values to reconstruct the right, middle, and left interface values. The grid dependent linear weights of the WENO are recovered by inverting a Vandermode-like linear system of equations with spatially varying coefficients. A scheme for calculating the linear we…
A new approach to the kinetic simulation of plasmas in complex geometries, based on the Particle-in- Cell (PIC) simulation method, is explored. In the two dimensional (2d) electrostatic version of our method, called the Arbitrary Curvilinear Coordinate PIC (ACC-PIC) method, all essential PIC operations are carried out in 2d on a uniform grid on the unit square logical domain, and mapped to a nonuniform boundary-fitted grid on the physical domain. As the resulting logical grid equations of motion are not separable, we have developed an extension of the semi-implicit Modified Leapfrog (ML) integ…
We present a new fourth-order, finite-volume hydrodynamics code named Apsara. The code employs a high-order, finite-volume method for mapped coordinates with extensions for nonlinear hyperbolic conservation laws. Apsara can handle arbitrary structured curvilinear meshes in three spatial dimensions. The code has successfully passed several hydrodynamic test problems, including the advection of a Gaussian density profile and a nonlinear vortex and the propagation of linear acoustic waves. For these test problems, Apsara produces fourth-order accurate results in case of smooth grid mappings. The …
We present a new high-order accurate Lagrangian discontinuous Galerkin (DG) hydrodynamic method to simulate material dynamics (for e.g., gasses, fluids, and solids) with up to fourth-order accuracy on cubic meshes. The variables, such as specific volume, velocity, specific total energy, and deformation gradient fields within a cell, are represented with a polynomial constructed from a novel hierarchical orthogonal basis about the center of mass, which decouples the moments of the solution because the mass matrix is diagonal. The discontinuity in the polynomials at the cell boundary is addresse…
High-Resolution Weighted Essentially Non-Oscillatory Compact Least-Squares Schemes with Implicit Time Integration for Compressible Navier-Stokes Equations on Curvilinear Grids
Ningbo University（宁波大学）；Northeast Normal University（东北师范大学）；Tsinghua University（清华大学）
Efficient simulation of plasmas in various contexts often involves the use of meshes that conform to the intrinsic geometry of the system under consideration. We present here a description of a new magnetohydrodynamic code, Gamera (Grid Agnostic MHD for Extended Research Applications), designed to combine geometric flexibility with high-order spatial reconstruction and constrained transport to maintain the divergence-free magnetic field. Gamera carries on the legacy of its predecessor, the LFM (Lyon-Fedder-Mobarry), a research code whose use in space physics has spanned three decades. At the t…
A high fidelity flow simulation for complex geometries for high Reynolds number ($Re$) flow is still very challenging, which requires more powerful computational capability of HPC system. However, the development of HPC with traditional CPU architecture suffers bottlenecks due to its high power consumption and technical difficulties. Heterogeneous architecture computation is raised to be a promising solution of difficulties of HPC development. GPU accelerating technology has been utilized in low order scheme CFD solvers on structured grid and high order scheme solvers on unstructured meshes. T…
We present a new method for numerical hydrodynamics which uses a multidimensional generalisation of the Roe solver and operates on an unstructured triangular mesh. The main advantage over traditional methods based on Riemann solvers, which commonly use one-dimensional flux estimates as building blocks for a multidimensional integration, is its inherently multidimensional nature, and as a consequence its ability to recognise multidimensional stationary states that are not hydrostatic. A second novelty is the focus on Graphics Processing Units (GPUs). By tailoring the algorithms specifically to …
The speed of sound in two-phase pipe flow systems is often several orders of magnitude greater than the travelling speed of hydraulic information (volume fractions.) Dynamically simulating such flows requires resolution of acoustic and hydraulic waves existing at vastly different spatial and temporal scales. If simulated on the same numerical grid, the need for accuracy in hydraulic waves will necessitate an exaggerated resolution of acoustic waves. Likewise, time steps restricted by the speed of sound are small compared to the time scales active in hydraulic waves. This constitutes a waste of…
Second order finite volume scheme for Euler equations with gravity which is well-balanced for general equations of state and grid systems
Univ. of Würzburg（维尔茨堡大学）；TIFR Center for Applicable Mathematics（TIFR应用数学中心）；Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik（海德堡大学天文学中心，理论天体物理研究所）；Heidelberger Institut für Theoretische Studien（海德堡理论研究所）
This article examines the use of characteristic methods in stratified two-phase pipe flow simulations for obtaining non-dissipative flow predictions. A Roe scheme and several methods based on the principle of characteristics are presented for the two-fluid model. The main focus is finding numerically efficient ways of capturing wave dynamics and flow regime transitions through direct simulation. Characteristic methods offer the possibility of simulating hyperbolic systems without numerical dissipation. These methods do however lack certain fundamental conservation properties. Challenges relate…
The design and implementation of a new framework for adaptive mesh refinement (AMR) calculations is described. It is intended primarily for applications in astrophysical fluid dynamics, but its flexible and modular design enables its use for a wide variety of physics. The framework works with both uniform and nonuniform grids in Cartesian and curvilinear coordinate systems. It adopts a dynamic execution model based on a simple design called a "task list" that improves parallel performance by overlapping communication and computation, simplifies the inclusion of a diverse range of physics, and …
We present a description of the adaptive mesh refinement (AMR) implementation of the PLUTO code for solving the equations of classical and special relativistic magnetohydrodynamics (MHD and RMHD). The current release exploits, in addition to the static grid version of the code, the distributed infrastructure of the CHOMBO library for multidimensional parallel computations over block-structured, adaptively refined grids. We employ a conservative finite-volume approach where primary flow quantities are discretized at the cell-center in a dimensionally unsplit fashion using the Corner Transport U…
