Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Error analysis of the Monte Carlo method for compressible magnetohydrodynamics

Eduard Feireisl, Maria Lukacova-Medvidova, Bangwei She, Yuhuan Yuan

Abstract

We study random compressible viscous magnetohydrodynamic flows. Combining the Monte Carlo method with a deterministic finite volume method we solve the random system numerically. Quantitative error estimates including statistical and deterministic errors are analyzed up to a stopping time of the exact solution. On the life-span of an exact strong solution we prove the convergence of the numerical solutions. Numerical experiments illustrate rich dynamics of random viscous compressible magnetohydrodynamics.

Error analysis of the Monte Carlo method for compressible magnetohydrodynamics

Abstract

We study random compressible viscous magnetohydrodynamic flows. Combining the Monte Carlo method with a deterministic finite volume method we solve the random system numerically. Quantitative error estimates including statistical and deterministic errors are analyzed up to a stopping time of the exact solution. On the life-span of an exact strong solution we prove the convergence of the numerical solutions. Numerical experiments illustrate rich dynamics of random viscous compressible magnetohydrodynamics.

Paper Structure

This paper contains 34 sections, 16 theorems, 175 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Compressible MHD system
  3. Strategy and main results
  4. Strong solutions to the compressible MHD system
  5. Local existence and conditional regularity
  6. Data dependence
  7. Energy equality, relative energy
  8. Solutions with random data
  9. Strong Law of Large Numbers applied up to a stopping time
  10. Strong Law of Large Numbers on the life-span
  11. Structure-preserving finite volume method
  12. Notations
  13. Mesh.
  14. Dual mesh.
  15. Function spaces.
  16. ...and 19 more sections

Key Result

Proposition 2.1

Let $Q \subset \mathbb{R}^3$ be a bounded domain of class $C^{k+1}$, $k \geq 3$. Suppose the initial data belong to the class and satisfy the compatibility conditions, Then there exists $0 < T_{\rm max} \leq \infty$ such that the compressible MHD system pde, with the boundary conditions bc, admits a strong solution $(\varrho, \mathbf{u}, {\bf B})$ in $[0, T_{\rm max}) \times Q$ unique in the cla

Figures (13)

  • Figure 1: Extended mesh in 2D.
  • Figure 2: Boundary cell $K$.
  • Figure 3: Sine wave problem. Deterministic FV solutions $(\varrho, m_1, B_1)_{h_{ref}}(\omega=0, T)$ with $h_{ref} = 2/320$ at $T=0.6$ and the time evolution of $\| {\rm div}_h {\bf B}_h\|_{L^{1}(Q)}(\omega=0, t)$ with $h = 2/(10 \cdot 2^i), i = 1,\dots,5$.
  • Figure 4: Random sine wave problem. FV solutions $(\varrho, m_1, B_1)_{h_{ref}}(\omega,T)$ (from top to bottom) with $h_{ref} = 2/320$ and $500$ samples at $T=0.6$. From left to right: mean, deviation, and mean and deviations along $x=y$.
  • Figure 5: Random sine wave problem. Statistical errors: $E_1$ (left), $E_2$ (right). In the legend notations $\nabla {\bf u},$$\nabla \times {\bf B}$ are used for the discrete operators $\nabla_\mathcal{E} \mathbf{u}_h,\, {\rm curl}_h {\bf B}_h,$ respectively.
  • ...and 8 more figures

Theorems & Definitions (42)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1: Local existence, smooth data
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4: Conditional regularity
  • Corollary 2.5
  • Remark 2.6
  • Remark 3.1
  • ...and 32 more