2nd order ODE of Josephson’s Junction

The system of two coupled non-linear 2^nd order differential equations that represent the Josephson’s Junction phenomenon on superconductors is presented here, as adapted from Hairer et al . 

 

The equations describe the electron-pair interactions between two superconducting elements that are separated by a thin insulator, under the influence of an applied current source. The obvious first step in attempting to numerically integrate these equations is to decouple it.

Coupled equations can be decoupled in a variety of ways. Perhaps the easiest one is the brute-force method, where in a few algebraic sleights are performed to eliminate one the variables in either equation. In the subject case, multiplying the second equation by "alpha" and adding it to the first to eliminate one of the 2^nd derivative variables. Subsequently, this equation can be substituted into the first equation to obtain the decoupled versions. 

The algebraic sleight works best with a limited set of equations (usually up to two at the most). As the number of equations increases, the algebraic decoupling process becomes rather cumbersome. An alternative is to use the matrix formulation instead. The Josephson equations then become:

 A matrix inversion can then be performed to obtain the decoupled equations: 

Since the matrix that requires the inversion is a constant, only a single inversion is required in the integration process.

And the two 2^nd order ODE’s can be considered as being successfully decoupled. Numerical integration is just a step away, once the 2^nd order ODE’s are converted to 1^st order ODE’s.

Plotted below is the numerical solution of the equations, performed using a Rosenbrock technique. The initial conditions and values for the constants from Hairer et al were used for the analysis. 

More on the Semi-Implicit method

The matrix formulation of the semi-implicit methods, as indicated earlier, has the advantage of avoiding an iterative procedure to converge at a solution. However, the reformulation introduces two additional complications – matrix inversion and Jacobian.

Matrix inversion can be performed by using the LU-decomposition followed by the LU-back substitution. The Jacobians can be either defined analytically or computed numerically. The analytical definitions are obviously preferred so as to avoid any numerical approximation errors with the computation, with the caveat that analytical definitions can become a little tedious to program.

Yet another generalized representation of the semi-implicit method can be stated as follows:





The above representation tends to look more like the generalized RK methods, with the obvious exception of the ‘A’ matrix and the Jacobians!