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A Characteristic Mapping Method with Source Terms: Applications to Ideal Magnetohydrodynamics

Xi-Yuan Yin, Philipp Krah, Jean-Christophe Nave, Kai Schneider

Abstract

This work introduces a generalized characteristic mapping method designed to handle non-linear advection with source terms. The semi-Lagrangian approach advances the flow map, incorporating the source term via the Duhamel integral. We derive a recursive formula for the time decomposition of the map and the source term integral, enhancing computational efficiency. Benchmark computations are presented for a test case with an exact solution and for two-dimensional ideal incompressible magnetohydrodynamics (MHD). Results demonstrate third-order accuracy in both space and time. The submap decomposition method achieves exceptionally high resolution, as illustrated by zooming into fine-scale current sheets. An error estimate is performed and suggests third order convergence in space and time.

A Characteristic Mapping Method with Source Terms: Applications to Ideal Magnetohydrodynamics

Abstract

This work introduces a generalized characteristic mapping method designed to handle non-linear advection with source terms. The semi-Lagrangian approach advances the flow map, incorporating the source term via the Duhamel integral. We derive a recursive formula for the time decomposition of the map and the source term integral, enhancing computational efficiency. Benchmark computations are presented for a test case with an exact solution and for two-dimensional ideal incompressible magnetohydrodynamics (MHD). Results demonstrate third-order accuracy in both space and time. The submap decomposition method achieves exceptionally high resolution, as illustrated by zooming into fine-scale current sheets. An error estimate is performed and suggests third order convergence in space and time.

Paper Structure

This paper contains 13 sections, 1 theorem, 45 equations, 8 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Characteristic Mapping Method
  3. Source Terms
  4. Ideal Magnetohydrodynamics in Vorticity Formulation
  5. Numerical Implementation
  6. Numerical Implementation of the Source Term
  7. Error Analysis
  8. Numerical Studies
  9. Manufactured solution: linear advection with source term
  10. Orszag--Tang test case
  11. Convergence in space and time
  12. Comparison with literature results
  13. Conclusion

Key Result

Proposition 1

The CMM for the ideal MHD equations in two dimensions using Hermite cubic spatial interpolation and $3^{rd}$ order Lagrange time extrapolation and Runge-Kutta 4 integration has a numerical error of order $\mathcal{O}(N^{-3} + {\Delta t}^{3} ) + o(l_1 + l_2)$ where $h$ and ${\Delta t}$ are the spatia

Figures (8)

  • Figure 1: Linear advection swirl test case source term analysis. Shown is the solution $\theta({\bm x}, t)$, subintegrals $F_{[t,\tau_i]}$, source term $f({\bm x}, t)$ and corresponding Fourier spectra (from left to right), for $t=0.4, 1, 1.6$ and 2 (from top to bottom).
  • Figure 2: Convergence in space ($\Delta_x$) and time ($\Delta_t$) for the swirl test case, compared to the analytical reference solution.
  • Figure 3: Time evolution of the vorticity $\omega$ (left) and current density $j$ (right) from top to bottom $t=0.1,0.4,0.9$ for OT.
  • Figure 4: Zoom with center $(x,y)=(L/2,L/2)$ of the current density $j$ at time $t=4.0$ for OT. The zoom factor between each image is 2.
  • Figure 5: Convergence in space (a) and time (b) for OT. Shown is the relative error with respect to the reference solution in the $L^\infty$ norm.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Proposition 1