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A high-order adaptive scheme for six-dimensional convection simulations

Author of the code: Erwan Deriaz
Publication introducing and exploiting the code:

Table of Contents


Figure 1: Refinement of a two dimensional grid: on the left the refinement process, on the right the resulting adaptive grid. From Deriaz (2015).

Even though computer power increases exponentially with time, resolving Vlasov-Poisson equations in full six-dimensional phase-space remains challenging nowadays and requires special computing techniques. Here we explore the use of Adaptive Mesh Refinement (AMR), which allows for a substantial gain in terms of memory and computational time.

In six dimensions, the operations should have a complexity independent of the number of dimensions. In order to avoid excessive dissipation of the solution, we use a third-order scheme.

Regarding refinement structure, we propose to use an adaptive point-centered dyadic grid, as illustrated by Figure 1.

Let us check the result of a straightforward recipe with the simplest numerical tools: Eulerian discretization, polynomial interpolation and finite differences for hyperbolic partial differential equations.

Interaction of two Plummer spheres in 3D-3V

Figure 2: Initial conditions for the interaction of two Plummer models. From Deriaz (2015).

Figure 3: Three dimensional view in physical space of the collision of two Plummer spheres (IFRIT 3D Data Visualization). From Deriaz (2015). Movie: anim40.mp4

Here, we perform a simple test, similarly as in Yoshikawa et al. (2013), consisting in simulating the collision between two Plummer spheres (Figure 2). When isolated, these systems should remain stable. Thanks to the AMR technique, we are able to simulate their interaction in full six-dimensional phase-space (Figures 3 and 4), with a reasonable level of accuracy (Figures 5 and 6) given the high dimensionality of the problem and the limited computational power we used for this test: this simulation involved 200 million sampling elements over 2,638 time steps and ran during 226 hours on a 12 core 2.66GHz Intel Xeon X5650. This means an average speed of about 50,000 points per second per thread.


We demonstrated the feasibility of simulations of Vlasov-Poisson systems in six-dimensional phase-space thanks to an innovative high-order adaptive scheme. It is characterized by a fourth-order interpolation in time, third-order in (phase-)space for the convection and second-order in space for the calculation of the force field.

Thanks to its weaker CFL condition compared to standard Eulerian schemes with uniform grids, it can compete with semi-Lagrangian solvers.

While the current version of the code is operational on shared memory architectures with a 80% efficient parallelization with OpenMP, it remains to be optimized to run on large configurations. The next steps are thus MPI parallelization in addition to the implementation of a fourth-order interpolet discretization in order to obtain substantial gains regarding the conservations shown on Figure 5.


Direct Integration of the Collisionless Boltzmann Equation in Six-dimensional Phase Space: Self-gravitating Systems, Yoshikawa K., Yoshida, N., Umemura, M., 2013, ApJ, 762, 116, 2013.

Figure 4: View in phase space of the collision of two Plummer spheres: a cut in the (z,w) plane (top panels), the projection onto the (z,w) plane (middle panels) and the corresponding adaptive grid (bottom panels). From Deriaz (2015).
top panels: acoupzwM.mp4
middle panels: projzwS.mp4
bottom panels: nrvpzwW.mp4

Figure 5: Conservation analysis. These are second-order estimations of the quantities. From Deriaz (2015).

Figure 6: Evolution of the refinement threshold parameter ε (normalized scale), the number of points (×108 for the scale), the kinetic energy Ec (normalized scale) and the maximum of the time derivative | ∂t f | (normalized scale). From Deriaz (2015).