Finding Solutions to a PDE System with Known Scalar Function

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SUMMARY

This discussion focuses on solving a system of partial differential equations (PDEs) for the vector field \(\vec{B}\) given the scalar function \(f\). The equations include \(\vec{B} \cdot \nabla f = 0\), \((\nabla \times \vec{B}) \cdot \nabla f = 0\), and \(\nabla \cdot \vec{B} = 0\). Participants agree that an analytic solution is unlikely and suggest exploring numerical methods, particularly the method of characteristics, to find solutions for \(\vec{B}\).

PREREQUISITES
  • Understanding of vector calculus and differential equations
  • Familiarity with the method of characteristics
  • Knowledge of numerical methods for PDEs
  • Experience with computational tools for solving PDEs, such as MATLAB or Python libraries
NEXT STEPS
  • Research the method of characteristics for solving PDEs
  • Explore numerical techniques for solving vector fields, such as finite difference or finite element methods
  • Learn about specific Python libraries for PDE solving, such as FEniCS or NumPy
  • Investigate existing literature on numerical solutions for divergence-free vector fields
USEFUL FOR

Mathematicians, physicists, and engineers involved in computational fluid dynamics or electromagnetic theory who are looking to solve complex PDE systems numerically.

vibe3
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Hi all, I am looking for ways to solve the following system of equations for [itex]\vec{B}[/itex]:
[tex] \vec{B} \cdot \nabla f = 0[/tex]
[tex] \left( \nabla \times \vec{B} \right) \cdot \nabla f = 0[/tex]
[tex] \nabla \cdot \vec{B} = 0[/tex]
and [itex]f[/itex] is a known scalar function. I think we can assume there is a solution since we have 3 equations and 3 unknown components of [itex]\vec{B}[/itex].

I don't think there is an analytic solution here, but could someone give some pointers on how one would solve this system numerically?
 
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Method of characteristics I think?
 

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