Exponentially fitted multisymplectic scheme for conservative Maxwell equations with oscillary solutions.

Area of Science:

• Computational Electromagnetics
• Numerical Analysis
• Applied Mathematics

Background:

• Maxwell's equations are fundamental in electromagnetism but complex to solve numerically, especially for oscillatory solutions.
• Existing numerical schemes may struggle with energy conservation and stability for these problems.
• Efficient and accurate methods are needed for simulating electromagnetic phenomena and parameter estimation.

Purpose of the Study:

• To develop a stable, conservative, and efficient numerical scheme for conservative Maxwell equations with periodic oscillatory solutions.
• To reduce computational cost by splitting the equations into local one-dimension (LOD) forms.
• To assess the scheme's performance in parameter inversion for electric permittivity.

Main Methods:

• Adoption of an exponentially fitted trapezoidal scheme, a type of multisymplectic scheme, for approximating temporal and spatial derivatives.
• Splitting Maxwell's equations into three local one-dimension (LOD) Maxwell equations.
• Proving unconditional stability and convergence of the LOD multisymplectic scheme and analyzing numerical dispersion relation.
• Applying the least squares method combined with the LOD multisymplectic scheme for parameter inversion.

Main Results:

• The proposed multisymplectic scheme satisfies two discrete energy conservation laws and preserves two discrete divergences.
• The LOD multisymplectic scheme is proven to be unconditionally stable and convergent.
• Numerical examples confirm the scheme's efficiency, stability, and conservation properties for oscillatory solutions.
• Parameter inversion results show good agreement between the fitted electric permittivity and measured data, demonstrating the method's effectiveness under small random errors.

Conclusions:

• The LOD multisymplectic scheme offers an efficient, stable, and conservative approach for solving conservative Maxwell equations with oscillatory solutions.
• The combination of the least squares method and the LOD multisymplectic scheme is effective for estimating model parameters, even with noisy data.
• This work provides a valuable numerical tool for computational electromagnetics and inverse problems.