Tracking Smarter, Not Harder: A Hybrid Characteristic Mapping Strategy for Vlasov–Poisson

We present a hybrid semi-Lagrangian method for the Vlasov–Poisson equation, combining the Characteristic Mapping Method (CMM)[1] with the Numerical Flow Iteration (NuFi)[2] scheme. Both approaches leverage the semigroup property of the underlying diffeomorphic flow to accurately trace characteristics back to their origins via a flow map, enabling precise evaluation of the solution at any point in phase space. NuFi constructs the flow map iteratively, preserving symplectic structure and conserving energy, but incurs quadratic computational cost with respect to time. However, it stores the map in a low-dimensional, implicit form dependent on the electric field. In contrast, CMM composes the flow map from explicitly stored submaps, achieving linear time complexity at the cost of higher memory usage. To combine their strengths, we use NuFi for local time stepping and decompose the resulting map into submaps composed through CMM. This strategy reduces overall complexity while maintaining accuracy and structure preservation. The key challenge, balancing memory requirements of CMM with the computational efficiency of the combined scheme, is addressed in the poster in detail.

[1] P Krah, XY Yin, J Bergmann, JC Nave and K Schneider. A characteristic mapping method for Vlasov-Poisson with extreme resolution properties. Comm. Comput. Phys., 35 (4), 905--937, 2024.

[2] M. Kirchhart and R.P. Wilhelm. The Numerical Flow Iteration for the Vlasov–Poisson Equation. SIAM J. Sci. Comput., 46(3), A1972-A1997, 2024.