## Vlasov Equations

### Project summary

- Project name: Splitting methods for the Vlasov-Poisson and Vlasov-Maxwell equations
- FWF project id: P25346
- University: University of Innsbruck, Department of Mathematics
- Field: numerical analysis, applied mathematics, plasma physics
- Keywords: abstract evolution equations, convergence analysis, splitting methods, high-order methods, discontinuous Galerkin, Vlasov-Maxwell equation

**Short project description**

The Vlasov-Poisson and Vlasov-Maxwell equations are the most fundamental description of a (collisionless) plasma. The equations describe the evolution of a particle-probability distribution in 3+3 dimensional phase space coupled to an electromagnetic field. The difficulties in obtaining a numerical solution of those equations are summarized in the following three statements:

- Due to the six-dimensional phase space the amount of memory required to store the interpolation is proportional to the sixth power of the number of grid points.
- The Vlasov equation is stiff (i.e. the time step size is limited by the CFL condition)
- The coupling to the Maxwell/Poisson equation makes the system highly non-linear.

A numerical scheme based on Strang splitting has been proposed that translates the basis functions of some interpolation space and projects the translated basis function back onto the proper subspace.

Various interpolation schemes for the above mentioned algorithm have also been investigated. It has been found that the discontinuous Galerkin method is extremely competitive performance wise, while most of its desirable features (such as locality) remain.

Therefore the aim of this project is to:

- supply an in-depth numerical analysis of Strang splitting for Vlasov-type equations;
- extend the achieved results to higher-order splitting methods;
- provide a convergence analysis of the fully discrete problems (using discontinuous Galerkin in space);
- extend the previous results to higher-order methods in space.

To implement higher-order splitting methods we have to evaluate the force term at certain intermediate steps. We will develop a strategy that leads to computationally efficient schemes.

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