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We extend a family of high-resolution, semidiscrete central schemes for hyperbolic systems of conservation laws to three-space dimensions. Details of the schemes, their implementation, and properties are presented together with results from several prototypical applications of hyperbolic conservation laws including a nonlinear scalar equation, the Euler equations of gas dynamics, and the ideal magnetohydrodynamic equations. Parallel scaling analysis and grid-independent results including contours and isosurfaces of density and velocity and magnetic field vectors are shown in this study, confirming the ability of these types of solvers to approximate the solutions of hyperbolic equations efficiently and accurately.

Over the past couple of decades, much work has gone into the construction, analysis, and implementation of modern numerical algorithms for the approximate solution of systems of nonlinear hyperbolic conservation laws of the form

Numerical solutions of these equations are of tremendous practical importance as they govern a variety of physical phenomena in natural and engineering applications. In particular, a number of high-resolution schemes have been developed and tested for this purpose [

These central schemes enjoy a major advantage of simplicity over methods like upwind schemes, in that no approximate Riemann solvers are involved in their construction. To provide a brief background, in 1990, Nessayahu and Tadmor introduced a fully discrete second-order nonoscillatory central scheme (NT scheme), [

The schemes presented in Kurganov and Tadmor’s study saw further developments in the form of third-order extensions [

Starting with a general hyperbolic conservation law in three-space dimensions, (

Modified central differencing in three-space dimensions.

Following [

Reconstructions in the

The cell interface values in the

They are calculated via a nonoscillatory piecewise polynomial reconstruction given by

The resulting semi-discrete scheme in the limit as

with numerical fluxes

for the third-order CWENO reconstruction without any diagonal smoothing (diagonal smoothing described in Section

This section provides a third-order nonoscillatory reconstruction in three-space dimensions, that was implemented for computing the solutions of hyperbolic conservation laws (see (

conserve the pair of cell averages

is determined so as to satisfy

Note that for the central parabola equation (

The conservation of the cell averages

In the case of systems of equations, the smoothness indicators are given by the

represents its

The interface values can now be calculated from (

Similar reconstructions to (

This section presents results from the solutions of a single scalar equation (

The presentation of three-dimensional semi-discrete schemes, formulated in Section

The equation is solved in a

and zero elsewhere.

Firstly, the solution of the previous equation is presented at various grids to assess the convergence of such a scheme. Also note that no diagonal smoothing is applied in this case. In Figure

Instantaneous solutions, at

Next, the schemes presented in this paper are compared to the second-order scheme of Kurganov and Tadmor [

Instantaneous solutions, at

Figure

Instantaneous solutions of the nonlinear scalar equation (

The computational requirements for the solution of hyperbolic problems could become prohibitive in the case of three-dimensional, geometrically complex enclosures. These requirements increase further when realistic fluid flows like viscous or turbulent flows are considered, thereby requiring larger computational effort and memory. Recent developments in high-performance computing promise a substantial increase in computational speed and offer new possibilities for more accurate simulations. Three-dimensional domain decomposition is used to speed the calculations, where the computational domain is decomposed into a number of rectangular blocks with each processor being responsible for a single block. An example of this decomposition can be seen by the gaps in the grid in Figure

Mesh and domain decomposition for

Most of the calculations in the interior of each of the subdomains are independent of the domain decomposition and can continue as if they are performed serially. Problems arise near the subdomain boundaries where, for example, finite differences calculated adjacent to the subdomain boundaries may need several points outside the subdomain. To support these circumstances, two rows of “ghost points” are carried along with the interior solutions that contain copies of the interior solution from the neighboring subdomain. These points are exchanged and updated from neighboring processors as needed to ensure that all near-wall calculations are performed with current variable values.

If a uniform grid is used, then the subdomains in each direction will contain equal number of grid points. However, for a nonuniform grid, the the division locations between the subdomains need to be selected to provide good load balancing or an equivalent amount of work for each processor in each time step. Hence, for the purpose of a scaling analysis, Figure

Parallel scaling analysis for the solution of the Euler hydrodynamics system (

Figure

Euler equations of gas dynamics are given by

Here

The evolution of the Richtmyer-Meshkov instability (RMI) [

The initial conditions were adapted from the Mach

Instantaneous contours of initial density across centerline

The following boundary conditions were used: (a) inflow at the test section entrance in the streamwise

Figures

Instantaneous contours of density across centerline

Instantaneous isosurfaces of density (

The system of equations for ideal magnetohydrodynamics (MHD) is given by

Here

The

with

Instantaneous contours of density across

A way of demonstrating the accuracy of a numerical method is to determine whether the solenoidal constraint

Instantaneous surface plot of the divergence of the magnetic field (

Isosurface of the density at a value

Instantaneous contours of density across

Instantaneous slices across the

Extensions of the semi-discrete schemes of Balbás and Tadmor [