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2.1. Operator-split equations of MHD. The full set of equations for multidimensional hydrodynamics naturally splits into a set of one-dimensional equations with no cross-coupling. Unfortunately, this statement is not true for the equations of MHD; the restriction V-B = 0 naturally induces some cross-coupling between the different directions. In this section, we present a set of one-dimensional equations for MHD that is suitable for operator-splitting. We begin with the full set of equations written as: The right-hand side of these equations differs from the conservation form of the equations by terms of the form V-B(...), where the terms inside the parentheses are not differentiated. Our reason for using this form was to minimize the extent to which various terms in the discrete evolution operators in each of the coordinate directions in the operator split algorithm would have to sum to zero due to the divergence-free constraint on the magnetic field. The use of this nonconservation form is similar to the common practice, in incompressible flow calculations, of using advective differencing of the velocity fields, rather than conservative differencing (see, for example, Bell, Colella, and Glaz [1]). Also, it has been shown by Brackbill and Barnes [4] that using the nonconservation form of the momentum equations reduces the effect of magnetic monopole forces on the dynamics of the system. In the above equations, we have used the following notation:

Figure 1 2.1. Operator-split equations of MHD. The full set of equations for multidimensional hydrodynamics naturally splits into a set of one-dimensional equations with no cross-coupling. Unfortunately, this statement is not true for the equations of MHD; the restriction V-B = 0 naturally induces some cross-coupling between the different directions. In this section, we present a set of one-dimensional equations for MHD that is suitable for operator-splitting. We begin with the full set of equations written as: The right-hand side of these equations differs from the conservation form of the equations by terms of the form V-B(...), where the terms inside the parentheses are not differentiated. Our reason for using this form was to minimize the extent to which various terms in the discrete evolution operators in each of the coordinate directions in the operator split algorithm would have to sum to zero due to the divergence-free constraint on the magnetic field. The use of this nonconservation form is similar to the common practice, in incompressible flow calculations, of using advective differencing of the velocity fields, rather than conservative differencing (see, for example, Bell, Colella, and Glaz [1]). Also, it has been shown by Brackbill and Barnes [4] that using the nonconservation form of the momentum equations reduces the effect of magnetic monopole forces on the dynamics of the system. In the above equations, we have used the following notation:

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With the equations written in this form, it is immediately apparent that a nonconservative, one-dimensional, operator-split set will be The one-dimensional equations are all of the form
With the equations written in this form, it is immediately apparent that a nonconservative, one-dimensional, operator-split set will be The one-dimensional equations are all of the form
We can use (24) as the starting point for systems. Given the approximate parameterization of the wave curves in §2 of BCT, we can define the Engquist~Osher flux for systems of equations as follows: and i;.(a) is the linear approximation to the wave speed given above.
We can use (24) as the starting point for systems. Given the approximate parameterization of the wave curves in §2 of BCT, we can define the Engquist~Osher flux for systems of equations as follows: and i;.(a) is the linear approximation to the wave speed given above.
As mentioned above, the Sod shock tube is a purely hydrodynamical Riemann problem in which the initial condition consists of two uniform states W,; and We separated by a discontinuity. The gas polytropic index is y = 1.4, and the initial conditions are 3.1. One-dimensional problems. Our suite of one-dimensional problems includes benchmarks commonly used to test numerical algorithms: the Sod shock tube problem [22], the Brio and Wu problem [5], and the strong version of the Sod shock tube problem [5]. In addition, we have studied the strong rarefaction problems considered by Einfeldt et al. [11]. Each problem tests a different aspect of the Riemann solver. The Sod shock tube problem tests the solver in the purely hydrodynamic limit, the Brio and Wu problem stresses our handling of linear degeneracies and compound waves, while the strong version of the Brio and Wu problem accents our handling of large-amplitude shock waves. Finally, the Einfeldt et al. problems test our ability to detect and handle strong rarefaction waves.
As mentioned above, the Sod shock tube is a purely hydrodynamical Riemann problem in which the initial condition consists of two uniform states W,; and We separated by a discontinuity. The gas polytropic index is y = 1.4, and the initial conditions are 3.1. One-dimensional problems. Our suite of one-dimensional problems includes benchmarks commonly used to test numerical algorithms: the Sod shock tube problem [22], the Brio and Wu problem [5], and the strong version of the Sod shock tube problem [5]. In addition, we have studied the strong rarefaction problems considered by Einfeldt et al. [11]. Each problem tests a different aspect of the Riemann solver. The Sod shock tube problem tests the solver in the purely hydrodynamic limit, the Brio and Wu problem stresses our handling of linear degeneracies and compound waves, while the strong version of the Brio and Wu problem accents our handling of large-amplitude shock waves. Finally, the Einfeldt et al. problems test our ability to detect and handle strong rarefaction waves.
From the analysis in [11], these initial conditions have a physical solution, but the solution is not linearizable. Our final one-dimensional test problem is drawn from the strong rarefaction problems discussed by Einfeldt et al. [11]. In our earlier discussion of the Riemann solver, we noted that conditions exist under which any linear Riemann will fail, even if the underlying physical prob- jem admits a solution. We have implemented a strong rarefaction wave detection algorithm that automatically applies additional dissipation whenever the Riemann solver fails. In Fig. 5 we show the results for the problem defined as 1 -2 —OQ—3 in [11]. This problem consists of two strong rarefaction waves moving symmetrically in opposite directions. With an adiabatic index y = 1.4, the initial conditions are
From the analysis in [11], these initial conditions have a physical solution, but the solution is not linearizable. Our final one-dimensional test problem is drawn from the strong rarefaction problems discussed by Einfeldt et al. [11]. In our earlier discussion of the Riemann solver, we noted that conditions exist under which any linear Riemann will fail, even if the underlying physical prob- jem admits a solution. We have implemented a strong rarefaction wave detection algorithm that automatically applies additional dissipation whenever the Riemann solver fails. In Fig. 5 we show the results for the problem defined as 1 -2 —OQ—3 in [11]. This problem consists of two strong rarefaction waves moving symmetrically in opposite directions. With an adiabatic index y = 1.4, the initial conditions are
Fic. 1. The solution to the Sod shock tube problem on a uniform mesh with 100 grid points is shown after 5¢ timesteps. The size of the timestep was controlled by the CFL condition with CFL number of 0.8.
Fic. 1. The solution to the Sod shock tube problem on a uniform mesh with 100 grid points is shown after 5¢ timesteps. The size of the timestep was controlled by the CFL condition with CFL number of 0.8.
Fic. 2. The comparison between the density profiles of the solution to the Sod shock tube problem computed after 50 timesteps by using different algorithms is presented. The solid line is the solution computed with the presen code; the dashed line is the solution computed with the PPM code in which the contact detection algorithm has beer turned off; the dashed-dotted line is the PPM solution with detection of contact discontinuities. The effect of contact detection on the resolution of the contact discontinuity is clearly visible.
Fic. 2. The comparison between the density profiles of the solution to the Sod shock tube problem computed after 50 timesteps by using different algorithms is presented. The solid line is the solution computed with the presen code; the dashed line is the solution computed with the PPM code in which the contact detection algorithm has beer turned off; the dashed-dotted line is the PPM solution with detection of contact discontinuities. The effect of contact detection on the resolution of the contact discontinuity is clearly visible.
Fic. 3. The solution to the Brio and Wu magnetized shock tube problem on a uniform mesh with 800 grid points is shown after 400 timesteps. The initial condition consists of a jump discontinuity in pressure and density separated by a current sheet.
Fic. 3. The solution to the Brio and Wu magnetized shock tube problem on a uniform mesh with 800 grid points is shown after 400 timesteps. The initial condition consists of a jump discontinuity in pressure and density separated by a current sheet.
Fic. 4. The solution to the strong version of the shock tube problem on a mesh with 200 grid points is shown after 100 timesteps. Here By = 0 and By, changes sign across the initial discontinuity. A CFL condition of 0.8 was used.
Fic. 4. The solution to the strong version of the shock tube problem on a mesh with 200 grid points is shown after 100 timesteps. Here By = 0 and By, changes sign across the initial discontinuity. A CFL condition of 0.8 was used.
Fic. 5. The solution to the Einfeldt et al. {11] 1-2-—-O~3 strong rarefaction problem on a mesh with 100 gria points is shown at time 0.1.
Fic. 5. The solution to the Einfeldt et al. {11] 1-2-—-O~3 strong rarefaction problem on a mesh with 100 gria points is shown at time 0.1.
Fic. 7. The solution to the spherically symmetric explosion problem with an initial field Byg = 10 ts show fler 48 timesteps. All the other conditions are the same as in Fig. 6. Fic. 6. The solution to the spherically symmetric explosion problem in the hydrodynamic limit Byo = 0 is shown on a mesh with 120-by-120 grid points after 48 timesteps. For each variable, there are 20 equally spaced contour levels. A CFL condition of 0.8 was used. No artificial viscosity or flattening was applied.
Fic. 7. The solution to the spherically symmetric explosion problem with an initial field Byg = 10 ts show fler 48 timesteps. All the other conditions are the same as in Fig. 6. Fic. 6. The solution to the spherically symmetric explosion problem in the hydrodynamic limit Byo = 0 is shown on a mesh with 120-by-120 grid points after 48 timesteps. For each variable, there are 20 equally spaced contour levels. A CFL condition of 0.8 was used. No artificial viscosity or flattening was applied.
Fic. 8. The solution to the spherically symmetric explosion problem with an initial field Byg = 100 is shown after 48 timesteps. All the other conditions are the same as in Fig. 6.
Fic. 8. The solution to the spherically symmetric explosion problem with an initial field Byg = 100 is shown after 48 timesteps. All the other conditions are the same as in Fig. 6.
Fic. 9. The evolution of the compressible Orszag—Tang vortex system for the case with initial Mach number Mo = 1 and initial B = 40 . The solution was computed on a periodic cartesian mesh with 192-by-192 zones ana is shown after 300 rimesteps (approximately 2 Alfven transit times). The upper plot shows equally spaced contour levels of the thermal pressure. There are several shock fronts that propagate at transverse directions with respect to the grid and eventually interact. The lower plot shows the pressure profile along the solid line at Y = 60. There are two sharp jumps clearly visible where the solid line intersects the two strong shock fronts.
Fic. 9. The evolution of the compressible Orszag—Tang vortex system for the case with initial Mach number Mo = 1 and initial B = 40 . The solution was computed on a periodic cartesian mesh with 192-by-192 zones ana is shown after 300 rimesteps (approximately 2 Alfven transit times). The upper plot shows equally spaced contour levels of the thermal pressure. There are several shock fronts that propagate at transverse directions with respect to the grid and eventually interact. The lower plot shows the pressure profile along the solid line at Y = 60. There are two sharp jumps clearly visible where the solid line intersects the two strong shock fronts.
Related topics:
MagnetohydrodynamicsCompressible Fluid Dynamics
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