Local cubic spline interpolation for Vlasov-type equations on a multi-patch geometry

TL;DR

This work develops a local cubic spline interpolation framework on multi-patch geometries for solving Vlasov-type advection problems with a backward semi-Lagrangian scheme. By enforcing a global-C1 continuity across patch interfaces through Hermite boundary conditions and a generalized derivative reconstruction, the method combines local patch splines with a small, partially global system to ensure accurate and stable interface treatment. The authors provide explicit formulas for uniform patches, analyze asymptotic behavior as patch resolution increases, and extend the approach to 2D domains with conforming, non-conforming, and T-joint geometries, including a diocotron-guided-center test. Numerical results demonstrate close agreement with global spline solutions when available and robust performance on non-conforming meshes and refined patches, highlighting the approach’s applicability to tokamak-like poloidal cross-sections and potential for efficient high-fidelity advection in complex geometries.

Abstract

We present a semi-Lagrangian method for the numerical resolution of Vlasov-type equations on multi-patch meshes. Following N. Crouseilles et al. [A parallel Vlasov solver based on local cubic spline interpolation on patches. Journal of Computational Physics (2009)], we employ a local cubic spline interpolation with Hermite boundary conditions between the patches. The derivative reconstruction is adapted to cope with non-uniform meshes as well as non-conforming situations. In the conforming case, there are no longer any constraints on the number of points for each patch; however, a small global system must now be solved. In that case, the local spline representations coincide with the corresponding global spline reconstruction. Alternatively, we can choose not to apply the global system and the derivatives can be approximated. The influence of the most distant points diminishes as the number of points per patch increases. For uniform per patch configurations, a study of the explicit and asymptotic behavior of this influence has been led. The method is validated using a two-dimensional guiding-center model with an O-point. All the numerical results are carried out in the Gyselalib++ library.

Figures (24)

• Figure 1: On the left, an example of global domain split into 10 patches. Mesh for the ITER tokamak based on data from SOLEDGE3X and generated by K. Obrejan (https://soledge3x.onrender.com, article SOLEDGE3X). At the center, a zoom on the core with a reference mapping with O-point. On the right, a zoom on the separatrix ($--$) intersection forming the X-point with a reference mapping.
• Figure 2: Logical and physical domains. We note $(r,\theta)$ the curvilinear coordinates on the logical domain and $(x,y)$ the Cartesian coordinates on the physical domain.
• Figure 3: Constant advection on a periodic domain split into $N_p$ patches. The function on patch $p$ is advected with a constant velocity $v<0$. The feet of the characteristics land in the patch $p$ and $p+1$. To determine the function at $t^{n+1}$, we interpolate the function representation at $t^{n}$ using the information from patch $p$, $p+1$ and $p+2$.
• Figure 4: Two Hermite polynomials: $P_L$ defined on $[x^L_{i-1}, x^L_{i}]$ and $P_R$ defined on $[x^R_{i}, x^R_{i+1}]$ with functions connected across patches between $x_i^L$ and $x_i^R$.
• Figure 5: Example of a 1D multi-patch of the global domain split into $N_p = 4$ patches. The spline representation is composed of $N_p = 4$ local spline representations. The relation \ref{['relation_interfaces']} can be given at the interfaces $\mathcal{X}_1$, $\mathcal{X}_2$ and $\mathcal{X}_3$ on the figure.