The discretization of the explicit part is extended to three spatial dimensions by using an unsplit finite volume method according to [
35], which writes
$$\begin{aligned} \begin{aligned} \Delta \textbf{q}_{i,j,k}^* = \Delta \textbf{q}_{i,j,k}^n&-\frac{{\Delta t}}{{\Delta x}} \left( \hat{\textbf{F}}_{i+1/2,j,k}^{c} - \hat{\textbf{F}}_{i-1/2,j,k}^{c}\right) + \frac{{\Delta t}}{{\Delta x}} \Bigl ( \textbf{F}^c(\tilde{\textbf{q}}_{i+1/2,j,k}) - \textbf{F}^c(\tilde{\textbf{q}}_{i-1/2,j,k}) \Bigr ) \\&-\frac{{\Delta t}}{{\Delta y}}\left( \hat{\textbf{G}}_{i,j+1/2,k}^{c} - \hat{\textbf{G}}_{i,j-1/2,k}^{c}\right) + \frac{{\Delta t}}{{\Delta y}} \Bigl ( \textbf{G}^c(\tilde{\textbf{q}}_{i,j+1/2,k}) - \textbf{G}^c(\tilde{\textbf{q}}_{i,j-1/2,k})\Bigr ) \\&-\frac{{\Delta t}}{{\Delta z}} \left( \hat{\textbf{H}}_{i,j,k+1/2}^{c} - \hat{\textbf{H}}_{i,j,k-1/2}^{c}\right) + \frac{{\Delta t}}{{\Delta z}} \Bigl (\textbf{H}^c(\tilde{\textbf{q}}_{i,j,k+1/2}) - \textbf{H}^c(\tilde{\textbf{q}}_{i,j,k-1/2})\Bigr ) \\&+ {\Delta t}\hat{\textbf{S}}_{i,j,k} - {\Delta t}\textbf{S}(\tilde{\textbf{q}}_{i,j,k}). \end{aligned} \nonumber \\ \end{aligned}$$
(39)
The numerical fluxes
\(\hat{\textbf{F}}^c\),
\(\hat{\textbf{G}}^c\) and
\(\hat{\textbf{H}}^c\) have the form of the Rusanov flux and are constructed as in (
29). For the source term
\(\hat{\textbf{S}}\) we again use the cell centered value. The updated density
\(\Delta \rho _{i,j,k}^{n+1}\) and magnetic field
\(\Delta \textbf{B}_{i,j,k}^{n+1}\) are equal to their explicit update, since in the splitting the complete flux of these components is explicit. The update of the momentum components, on the other hand, contains implicit parts and reads in the fully discrete and three-dimensional form as
$$\begin{aligned} (\Delta \rho u)_{i,j,k}^{n+1}&=(\Delta \rho u)_{i,j,k}^{*} - \frac{{\Delta t}}{2{\Delta x}}\left( p_{i+1,j,k}^{n+1} - \tilde{p}_{i+1,j,k} - p_{i-1,j,k}^{n+1} + \tilde{p}_{i-1,j,k} \right) , \end{aligned}$$
(40a)
$$\begin{aligned} (\Delta \rho v)_{i,j,k}^{n+1}&=(\Delta \rho v)_{i,j,k}^{*} - \frac{{\Delta t}}{2{\Delta y}}\left( p_{i,j+1,k}^{n+1} - \tilde{p}_{i,j+1,k} - p_{i,j-1,k}^{n+1} + \tilde{p}_{i,j-1,k} \right) , \end{aligned}$$
(40b)
$$\begin{aligned} (\Delta \rho w)_{i,j,k}^{n+1}&=(\Delta \rho w)_{i,j,k}^{*} - \frac{{\Delta t}}{2{\Delta z}}\left( p_{i,j,k+1}^{n+1} - \tilde{p}_{i,j,k+1} - p_{i,j,k-1}^{n+1} + \tilde{p}_{i,j,k-1} \right) . \end{aligned}$$
(40c)
Implicit terms also appear in the update of the total energy:
$$\begin{aligned} \begin{aligned} (\Delta \rho E)_{i,j,k}^{n+1} = (\Delta \rho E)_{i,j,k}^*&- \frac{{\Delta t}}{2{\Delta x}} \left( h_{i+1,j,k}^n(\rho u)_{i+1,j,k}^{n+1} - h_{i-1,j,k}^n(\rho u)_{i-1,j,k}^{n+1} \right) \\&- \frac{{\Delta t}}{2{\Delta y}} \left( h_{i,j+1,k}^n(\rho v)_{i,j+1,k}^{n+1} - h_{i,j-1,k}^n(\rho v)_{i,j-1,k}^{n+1} \right) \\&- \frac{{\Delta t}}{2{\Delta z}} \left( h_{i,j,k+1}^n(\rho w)_{i,j,k+1}^{n+1} - h_{i,j,k-1}^n(\rho w)_{i,j,k-1}^{n+1} \right) . \end{aligned} \end{aligned}$$
(41)
The pressure
\(p^{n+1}\), which is needed for the updates (
40) and (
41), can be determined by solving the following elliptic equation:
$$\begin{aligned}&\frac{p_{i,j,k}^{n+1}}{\gamma -1} - \frac{{\Delta t}}{2{\Delta x}} \frac{(\rho u)_{i,j,k}^{n}}{2 \rho _{i,j,k}^{n+1}} \left( p_{i+1,j,k}^{n+1} - \tilde{p}_{i+1,j,k} - p_{i-1,j,k}^{n+1} + \tilde{p}_{i-1,j,k} \right) \nonumber \\&\qquad - \frac{{\Delta t}}{2{\Delta y}} \frac{(\rho v)_{i,j,k}^{n}}{2 \rho _{i,j,k}^{n+1}} \left( p_{i,j+1,k}^{n+1} - \tilde{p}_{i,j+1,k} - p_{i,j-1,k}^{n+1} + \tilde{p}_{i,j-1,k} \right) \nonumber \\&\qquad - \frac{{\Delta t}}{2{\Delta z}} \frac{(\rho w)_{i,j,k}^{n}}{2 \rho _{i,j,k}^{n+1}} \left( p_{i,j,k+1}^{n+1} - \tilde{p}_{i,j,k+1} - p_{i,j,k-1}^{n+1} + \tilde{p}_{i,j,k-1} \right) \nonumber \\&\qquad - \frac{{\Delta t}^2}{{\Delta x}^2} \Bigg [ \left( \frac{3}{4} h_{i-1,j,k}^n + \frac{1}{4} h_{i+1,j,k}^n \right) \left( p_{i-1,j,k}^{n+1}-\tilde{p}_{i-1,j,k}\right) \nonumber \\&\qquad - \left( h_{i-1,j,k}^n+h_{i+1,j,k}^n \right) \left( p_{i,j,k}^{n+1}-\tilde{p}_{i,j,k}\right) + \left( \frac{1}{4} h_{i-1,j,k}^n+\frac{3}{4} h_{i+1,j,k}^n \right) \left( p_{i+1,j,k}^{n+1}-\tilde{p}_{i+1,j,k} \right) \Bigg ]\nonumber \\&\qquad - \frac{{\Delta t}^2}{{\Delta y}^2} \Bigg [ \left( \frac{3}{4} h_{i,j-1,k}^n + \frac{1}{4} h_{i,j+1,k}^n \right) \left( p_{i,j-1,k}^{n+1}-\tilde{p}_{i,j-1,k}\right) \nonumber \\&\qquad - \left( h_{i,j-1,k}^n+h_{i,j+1,k}^n \right) \left( p_{i,j,k}^{n+1}-\tilde{p}_{i,j,k}\right) + \left( \frac{1}{4} h_{i,j-1,k}^n+\frac{3}{4} h_{i,j+1,k}^n \right) \left( p_{i,j+1,k}^{n+1}-\tilde{p}_{i,j+1,k}\right) \Bigg ] \nonumber \\&\qquad - \frac{{\Delta t}^2}{{\Delta z}^2} \Bigg [ \left( \frac{3}{4} h_{i,j,k-1}^n + \frac{1}{4} h_{i,j,k+1}^n \right) \left( p_{i,j,k-1}^{n+1}-\tilde{p}_{i,j,k-1}\right) \nonumber \\&\qquad - \left( h_{i,j,k-1}^n+h_{i,j,k+1}^n \right) \left( p_{i,j,k}^{n+1}-\tilde{p}_{i,j,k}\right) + \left( \frac{1}{4} h_{i,j,k-1}^n+\frac{3}{4} h_{i,j,k+1}^n \right) \left( p_{i,j,k+1}^{n+1}-\tilde{p}_{i,j,k+1}\right) \Bigg ] \nonumber \\&\quad = b_{i,j,k}^n \end{aligned}$$
(42)
with the right-hand side
$$\begin{aligned} \begin{aligned} b_{i,j,k}^n =&\frac{\tilde{p}_{i,j,k}}{\gamma -1} + (\Delta \rho E)_{i,j,k}^* - \Delta m_{i,j,k}^{n+1} \\&-\frac{(\rho u)_{i,j,k}^n}{2\rho _{i,j,k}^{n+1}} (\Delta \rho u)_{i,j,k}^* -\frac{{\Delta t}}{2{\Delta x}} \left( h_{i+1,j,k}^n (\Delta \rho u)_{i+1,j,k}^* - h_{i-1,j,k}^n (\Delta \rho u)_{i-1,j,k}^* \right) \\&-\frac{(\rho v)_{i,j,k}^n}{2\rho _{i,j,k}^{n+1}} (\Delta \rho v)_{i,j,k}^* -\frac{{\Delta t}}{2{\Delta y}} \left( h_{i,j+1,k}^n (\Delta \rho v)_{i,j+1,k}^* - h_{i,j-1,k}^n (\Delta \rho v)_{i,j-1,k}^* \right) \\&- \frac{(\rho w)_{i,j,k}^n}{2\rho _{i,j,k}^{n+1}} (\Delta \rho w)_{i,j,k}^* -\frac{{\Delta t}}{2{\Delta z}} \left( h_{i,j,k+1}^n (\Delta \rho w)_{i,j,k+1}^* - h_{i,j,k-1}^n (\Delta \rho w)_{i,j,k-1}^* \right) . \end{aligned} \end{aligned}$$
(43)