Asymptotically stable particle-in-cell methods for the Vlasov-Poisson system with a strong external magnetic field (Q2801781)

The authors present numerical schemes for the resolution of the Vlasov-Poisson system written as \(\varepsilon\frac{\partial f}{\partial t}+v_{\bot}\cdot \nabla _{x_{\bot }}f+(E(t,x_{\bot })+\frac{1}{\varepsilon } v_{\bot }\wedge B_{ext}(t,x_{\bot }))\cdot \nabla _{v_{\bot }}f=0\), \( E=-\nabla _{x_{\bot }}\phi \), \(\Delta _{x_{\bot }}\phi =\rho \). Here, \( f(x,v)\in \mathbb{R}^{d}\times \mathbb{R}^{d}\), \(d=1,2,3\), is the distribution of particles in phase space and \(\frac{1}{\varepsilon }B_{ext}\) is the external magnetic field. The trajectories of the particles can be expressed as the solution of the system \(\varepsilon \frac{dX}{dt}=V\), \( \varepsilon \frac{dV}{dt}=\frac{1}{\varepsilon }V\wedge B_{ext}(t,X)+E(t,X)\) with the initial conditions \(X(t^{0})=x_{\bot }^{0}\), \(V(t^{0})=v_{\bot }^{0} \). The numerical schemes whch are here proposed are particle-in-cell methods obtained when introducing time steps \(t^{n}=n\Delta t\) and considering the problem \(\varepsilon \frac{dX_{k}}{dt}=V_{k}\), \(\varepsilon \frac{dV_{k}}{dt} =\frac{1}{\varepsilon }V_{k}\wedge B_{ext}(t,X_{k})+E(t,X_{k})\) in the interval \(\left[ t^{n},t^{n+1}\right] \) with the initial conditions \( X_{k}(t^{n})=x_{k}^{n}\), \(V_{k}(t^{n})=v_{k}^{n}\). The authors introduce three types of semi-implicit schemes: first-order, second-order and third-order Runge-Kutta. For all these schemes, they prove consistency properties in the limit \(\varepsilon \rightarrow 0\) and for fixed \(\Delta t\) and for the first-order and some second-order schemes they further prove uniform consistency properties. The paper ends with the presentation of numerical simulations.