Maxwell equations

PDEs

There are no laws of physics that we use here -- because there is no derivation of the equations. Maxwell equations have long been written in terms of differential forms so I just need to interpret them as discrete ones, in order to derive a simulation.

The "differential forms" of the equations, i.e., the PDEs:

$\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}$ -- Faraday's Law

$\nabla \times \mathbf{H} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{D}}{\partial t}$ -- Ampere's Law

$\nabla \cdot \mathbf{D} = \frac{\rho}{\varepsilon_0}$ -- Gauss' Law

$\nabla \cdot \mathbf{B} = 0$ -- Gauss's Law for Magnetism

These letters stand for the following vector fields:

• $E$: electric field, also called • the electric field intensity;
• $B$: magnetic field, also called • the magnetic induction• the magnetic field density, • the magnetic flux density;
• $D$: electric displacement field, also called • the electric induction, • the electric flux density;
• $H$: magnetizing field, also called • auxiliary magnetic field, • magnetic field intensity, • magnetic field;
• $J$: total current density (including both free and bound current);

as well as these constants:

• $ε_0$: permittivity of free space, also called the electric constant, a universal constant;
• $μ_0$: permeability of free space, also called the magnetic constant, a universal constant.

Differential forms

Suppose this is happening on a manifold $M$. Then we can reconsider these physical quantities as differential forms. They are defined on its tangent bundle $TM$ and, dually, on the cotangent bundle $T^*M$.

Now, reconsidered as differential forms:

• $E$: 1-form on $TM$ or the primal complex in the discrete case,
• $B$: 2-form on $TM$ or on the primal complex in the discrete case,
• $D$: 2-form on $T^*M$ or the the dual complex in the discrete case,
• $H$: 1-form on $T^*M$ or the the dual complex in the discrete case,
• $J$: 2-form on $T^*M$ or the the dual complex in the discrete case.

They are also $0$-forms with respect to time.

For these forms:

$\oint_P E = -\frac{d}{dt} \int_A B,$

$\oint_P H = \frac{d}{dt} \int_A D + \int_A J,$

$\oint_S D = \int_V \rho,$

$\oint_S B = 0.$

Here $A$ is a surface bounded by a closed path $P$, $V$ is a solid bounded by surface $S$.

Equations of differential forms

Faraday’s Law applies to the tangent bundle or the primal complex in the discrete case: $$d E dt= -d_t B,$$ and Ampere’s Law to the cotangent bundle or the dual complex: $$d^* H dt = \mu_0\varepsilon_0 d_t D + \mu_0 J dt.$$ Here:

• $d$ and $d^*$ are the exterior derivative of the primal and the dual complex respectively (they are adjoint operators),
• $d_t$ is the exterior derivative with respect to time == the forward difference == difference of values of the function,
• $dt$ is the basic 1-form with respect to time,
• the product of forms is, in fact, the exterior product.

The rest of the laws are: $$d^* D = \frac{\rho}{\varepsilon_0},$$ $$d B = 0.$$

In terms of the Hodge star operator, the constitutive relations are: $$D=\varepsilon_0 \star E,$$ $$B=\mu_0 \star H.$$

If we think of these differential forms as discrete differential forms, then, in particular, the discrete exterior derivative $d$ is given by the transpose of the matrix of the boundary operator $\partial$, i.e., the incidence matrix (dimension 1) of the complex, and that of the dual is its transpose: $d^* = d^T =\partial.$

Note: one can use the same (4-dimensional) exterior derivative $d$ for both time and space if we assume that these 1- and 2-forms are "space only".

Further: $$F_{Hooke}= (\delta kdu)(x),$$ where $\delta$ is the codifferential.

The matrices of the codifferentials are $$\delta_t = \frac{1}{\Delta t^2}d^*_t,$$ $$\delta = \frac{1}{\Delta x^2}d^*,$$ where $\Delta t,\Delta x$ are the grid sizes.

See also Wave equation from Maxwell equations.

Simulation

From the above equations, we derive a simulation of the development of the electric field $E$:

Given:

• the current value of $E$,
• the current value of $B$,
• the increment of time $\Delta t$,
• the cell complex in the form of its boundary operator $\partial$, and its (incidence) matrices for dimensions 0,1,2,
• the matrices of the exterior derivative $d$ as the transposes of these matrices of $\partial$,
• the matrices of the exterior derivative $d$ for the dual complex equal to these matrices of $\partial$,
• the matrices of the Hodge star operator $\star$ (it's diagonal) and its inverse $\star ^{-1}$.

Compute:

• the exterior derivative $Q:=d E$ of $E$, then
  • this is the increment $Q = d_t B$ of $B$, so
  • the new value of $B$ is $B := B + Q \Delta t$,
• the Hodge dual $H := \star B / \mu_0$ of $B$,
• the exterior derivative $P := d^T H$ of $H$ (dual complex), then
  • this is the increment $R := d_t D = (P -\mu_0 J )/(\mu_0\varepsilon_0)$ of $D$, so
  • the new value of $D$ is $D := D + R \Delta t$,
• the Hodge dual $\star D / \varepsilon_0$ of $D$,
  • that's the new value of $E$.

More details needed...

Or, this "time-stepping": $$dE(t_n) = - \frac{B(t_{n+1/2}) - B(t_{n-1/2})}{\tau _n},$$ $$dH(t_{n+1/2}) = \frac{D(t_{n+1}) - D(t_{n-1}} {\tau _{n+1/2}} + J(t_{n+1/2}).$$

Note: The electric and magnetic fields may be jointly described by a 2-form $F$ in a 4-dimensional space-time manifold: $$d F = 0,$$ $$d \star F = J.$$