Appendix B: Modified leapfrog

Consider first the synchronized solutions, ${\vec x}_{_n}$, ${\vec v}_{_n}$, for positions and velocities, respectively, at the cycle n of time integration (instant t_n). Suppose that the old leapfrog-positions ${\vec x}_{_{n-1/2}}$, reached from the instant t_n to instant t_n-1/2, is receding by a half time-step $0.5\tau_{_{n-1/2}}$, where $\tau_{_{n-1/2}}$is a given time-step to advance positions from time-level n-1/2 to n+1/2. This time-step was estimated at time-level n-1/2, obeying some stability condition (e.g. Hilbert-Courant instability conditions).

From the above considerations, we have the following second-order approximation for positions at time-level n from time-level n-1/2:

$${\vec x}_{_n} \approx {\vec x}_{_{n-1/2}} +{1\over 2}\tau_{_{... ...{\vec v}_{_n} -{1\over 8}\tau_{_{n-1/2}}^2{\vec a}_{_{n-1/2}},$$ (B1)

where ${\vec a}_{_{n-1/2}}$ is the particle acceleration which depends in general on positions ( ${\vec x}_{_{n-1/2}}$) and position-synchronized velocities ( ${\vec v}_{_n-1/2}$).

Second, consider the final leapfrog-positions, ${\vec x}_{_{n+1/2}}$, obtained from the initial solution ${\vec x}_{_n}$ advancing from time-level n to n+1/2 by a half time-step, $\tau_{_{n+1/2}}$, yielded at time-level n+1/2:

$${\vec x}_{_{n+1/2}} \approx {\vec x}_{_n} +{1\over 2}\tau_{_{... ...{\vec v}_{_n} +{1\over 8}\tau_{_{n+1/2}}^2{\vec a}_{_{n-1/2}}.$$ (B2)

Adding the respective members of Eqs. (B1) and (B2), and then simplifying, we have the modified second-order leapfrog scheme for positions:

$${\vec x}_{_{n+1/2}} \approx {\vec x}_{_{n-1/2}} +\bar\tau{\vec v}_{_n} +\bar\tau\delta\tau{\vec a}_{_{n-1/2}},$$ (B3)

where

$$\bar\tau\equiv 0.5(\tau_{_{n+1/2}}+\tau_{_{n-1/2}}),$$ (B4)

and

$$\delta\tau\equiv 0.25(\tau_{_{n+1/2}}-\tau_{_{n-1/2}}).$$ (B5)

Velocities are integrated from time-level n to n+1, which is divided into the two steps: integrating with the same half time-step, $0.5\tau_{_{n+1/2}}$, from n to n+1/2, and from n+1/2 to n+1. After some algebra, this procedure results in the following second-order leapfrog-scheme for velocities:

$${\vec v}_{_{n+1}}= {\vec v}_{_n} +\tau_{_{n+1/2}}{\vec a}_{_{n+1/2}}.$$ (B6)

Time centered velocities may be estimated with the following first-order approximation:

$${\vec v}_{_{n+1/2}}= 0.5({\vec v}_{_{n+1}}+{\vec v}_{_n}),$$ (B7)

which propagates third-order error to the overall leapfrog fluid equations.

Thermal energies, or any other physical quantity obeying a first order differential equation of motion (e.g., magnetic field), may be advanced with an implicit second order scheme. Since specific thermal energy rate, $\dot u_i$, depends implicitly on particle's position and velocity, ( ${\vec x}_i$, ${\vec v}_i$), it is quite convenient to synchronize thermal energies with positions. Velocity terms in quantities like the velocity-divergence, artificial viscosity etc. shall be time-centered via Eq. (B7).

Considering the initial solutions for the specific thermal energy and specific thermal energy rate to be u_n-1/2 and $\dot u_{_{n-1/2}}$ respectively. Time-centered solutions may be obtained by integrating from time-level n-1/2 to n, and receding the final solutions from time-level n+1/2 to n. Both solutions can be written in a second order approximation:

$$u_{_n}\approx u_{_{n-1/2}} +{1\over 2}\tau_{_{n-1/2}}\dot u_{_{n-1/2}} +{1\over 8}\tau_{_{n-1/2}}^2\ddot u_{_n},$$ (B8)

and

$$u_{_n}\approx u_{_{n+1/2}} -{1\over 2}\tau_{_{n+1/2}}\dot u_{_{n+1/2}} +{1\over 8}\tau_{_{n+1/2}}^2\ddot u_{_n}.$$ (B9)

Subtracting the respective members of Eq. (B8) from Eq. (B9), we find

$$u_{_{n+1/2}}= u_{_{n-1/2}} +{1\over 2} \biggl\{ \tau_{_{n+1/2}}\dot u_{_{n+1/2}} +\tau_{_{n-1/2}}\dot u_{_{n-1/2}}$$

$$\;\;\;\;\;\; -{1\over 4}(\tau_{_{n+1/2}}^2-\tau_{_{n-1/2}}^2)\ddot u_{_{n}} \biggr\}\cdot$$ (B10)

Analogously, we find $\dot u_{_n}\approx\dot u_{_{n-1/2}}+0.5\tau_{_{n-1/2}}\ddot u_{_{n}}$ and $\dot u_{_n}\approx\dot u_{_{n+1/2}}+{1\over 2}\tau_{_{n+1/2}}\ddot u_{_{n}}$, from which we may extract the second derivative:

$$\ddot u_{_n}={\dot u_{_{n+1/2}}-\dot u_{_{n-1/2}}\over0.5(\tau_{_{n+1/2}}+\tau_{_{n-1/2}})}.$$ (B11)

Substituting $\ddot u_{_n}$ from Eq. (B11) in Eq. (B10), and after some simplifications, we have the following difference-scheme:

$$u_{_{n{+}1/2}}{=} u_{_{n{-}1/2}} {+}0.25(\tau_{_{n{+}1/2}}{+}\tau_{_{n{-}1/2}})(\dot u_{_{n{+}1/2}}{+}\dot u_{_{n{-}1/2}}).$$ (B12)