The Vlasov–Ampere system describes the evolution of a collisionless plasma where particles interact through self-consistent electromagnetic fields. This is a fundamental model in plasma physics used to study phenomena like Landau damping.
The system couples two key equations:
Vlasov equation: Describes how the distribution function \(f(t,x,\upsilon)\) (the probability density of finding particles at position \(x\) with velocity \(\upsilon\) at time \(t\)) evolves under the influence of an electric field.
Ampere’s law: Relates the electric field to the current density, which is determined by integrating the distribution function weighted by velocity.
Vlasov Equation
\[ \frac{\partial f}{\partial t} + \upsilon \frac{\partial f}{\partial x} - E(t,x) \frac{\partial f}{\partial \upsilon} = 0 \]
This equation expresses conservation: particles follow characteristics in phase space (positions in \(x\)-\(\upsilon\) space).
Ampere’s Law
\[ \frac{\partial E}{\partial t} = - J = - \int f\upsilon \; d\upsilon \]
Algorithm: Strang Splitting Approach
We solve the Vlasov–Ampere system using a Strang splitting method combined with the Fourier-Hermite method. The algorithm splits the problem into two advection steps:
1. Fourier Transform in Space
For each velocity point \(\upsilon_j\), compute the discrete Fourier transform of \(f^n(x_i,\upsilon_j)\): \[\hat{f}_k^n(\upsilon_j) = \sum_{i=0}^{n_x-1} f^n(x_i, \upsilon_j) e^{-2\pi i k x_i/L}\]
This transforms the spatial domain to the Fourier (wavenumber) domain, where advection becomes multiplication.
2. Velocity Advection (Half Step)
For each Fourier mode \(k \neq 0\): \[\hat{f}^{n+1}_k(\upsilon_j) = e^{-2i\pi k \upsilon_j \Delta t/L} \hat{f}^n_k(\upsilon_j)\]
Physical Meaning: In Fourier space, spatial derivatives become multiplications by \(ik\), so the advection \(\upsilon \frac{\partial f}{\partial x}\) becomes a phase shift proportional to \(k\upsilon\).
3. Compute Charge Density
From the distribution function in Fourier space: \[\rho_k^{n+1} = \Delta \upsilon \sum_j \hat{f}^{n+1}_k(\upsilon_j)\]
This integrates over all velocities to get the charge density for each mode.
4. Compute Electric Field
Using Poisson’s equation in Fourier space: \[\hat{E}^{n+1}_k = \frac{\rho^{n+1}_k L}{2i\pi k \epsilon_0} \quad \text{for } k \neq 0\]
In Fourier space, the Poisson equation \(\frac{\partial^2 \phi}{\partial x^2} = -\rho/\epsilon_0\) becomes algebraic: \(-k^2 \hat{\phi}_k = -\hat{\rho}_k/\epsilon_0\).
5. Handle Zero Mode
For \(k = 0\) (the average field): \[\hat{f}_{n+1}(\upsilon_j) = \hat{f}^n_k(\upsilon_j), \quad \hat{E}^{n+1}_k = \hat{E}^n_k\]
The zero mode is not updated because the average electric field is fixed (no net current in our periodic domain).
6. Space and Field Advection (Full Step)
Return to physical space by taking the inverse Fourier transform and apply the field’s influence on velocity distribution.
7. Inverse Fourier Transform
Transform both \(\hat{E}^{n+1}_k\) and \(\hat{f}^{n+1}_k(\upsilon_j)\) back to physical space to get \(E^{n+1}(x)\) and \(f^{n+1}(x,\upsilon)\).