SemiLagrangian.jl Documentation
Let us consider an abstract scalar advection equation of the form
$ \frac{∂f}{∂t}+ a(x, t) ⋅ ∇f = 0. $
The characteristic curves associated to this equation are the solutions of the ordinary differential equations
$ \frac{dX}{dt} = a(X(t), t) $
We shall denote by $X(t, x, s)$ the unique solution of this equation associated to the initial condition $X(s) = x$.
The classical semi-Lagrangian method is based on a backtracking of characteristics. Two steps are needed to update the distribution function $f^{n+1}$ at $t^{n+1}$ from its value $f^n$ at time $t^n$ :
- For each grid point $x_i$ compute $X(t^n; x_i, t^{n+1})$ the value of the characteristic at $t^n$ which takes the value $x_i$ at $t^{n+1}$.
- As the distribution solution of first equation verifies $f^{n+1}(x_i) = f^n(X(t^n; x_i, t^{n+1})),$ we obtain the desired value of $f^{n+1}(x_i)$ by computing $f^n(X(t^n;x_i,t^{n+1})$ by interpolation as $X(t^n; x_i, t^{n+1})$ is in general not a grid point.
Eric Sonnendrücker - Numerical methods for the Vlasov equations
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