Central finite volume methods with constrained transport divergence treatment for ideal MHD

Central finite volume methods for ideal and shallow water magnetohydrodynamics

“…Our CTCS method conserves the accuracy of the base scheme since both the temporal and the spatial integration of the induction equation are approximated using second-order quadrature rules. We have solved several classical two-dimensional ideal MHD/SMHD problems and compared our results with those we obtain when we apply the numerical method previously introduced in [18], where the dual cells are diamond shaped. Results obtained using both CTCS methods compare very well with one another and are in a very good agreement with the corresponding results in the literature, thus confirming the potential and efficiency of our methods.…”

“…The maximum absolute values of the divergence of the magnetic field in the numerical solution observed for the problems considered in this paper are within the 10 −11 -10 −14 range. Both numerical schemes (the diamond and the Cartesian dual cell schemes) are numerically equivalent; however the numerical method presented in this paper along with its corresponding CTCS procedure is simpler and easier to implement than the method presented in [18]; furthermore, the present numerical approach for solving ideal MHD problems saves about 25-30% of the computing time as compared with the diamond dual cell scheme and still achieves very competitive results with a very good agreement with those obtained using the diamond dual cell method, and with existing results in the literature. The proposed scheme is second-order accurate both in space and time thanks to piecewise linear interpolants and temporal quadrature rules of order two.…”

“…In a previous paper [18], we have presented a two-dimensional extension of the NT scheme where the original cells for the solution at time t n are Cartesian, while at time t n+1 , the solution is computed on the staggered diamond dual cells as is shown in Figure 2. A brief discussion of the method follows.…”

“…For some of the experiments we shall present a comparison of the results obtained using the purely Cartesian two-dimensional numerical method with those we have previously obtained using the diamond dual cell scheme presented in [18].…”

Unstaggered central schemes with constrained transport treatment for ideal and shallow water magnetohydrodynamics

“…More precisely, if the numerical solution obtained using an NT-type base scheme (at time t n+1 ) requires additional treatment in order to satisfy a physical property, a synchronization problem arises since any treatment of the updated solution usually requires the solution values computed at different previous times (e.g., at time t n , t n−1 and maybe earlier). The situation gets even harder when the control cells of the original and the staggered grids are not of the same shape/type [3,24]. In 1998, Jiang et al [17] presented a first unstaggered adaptation of the NT scheme; the method they proposed utilizes both iteration formulas of the original Nessyahu and Tadmor scheme (Eqs.…”