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Adaptive Grid Method

Automatic refinement in the grid size, where the field gradient is above a certain threshold value. This avoid a cpu intensive calculation of grid points, which are empty.

Cranc-Nicholson Method

Semi-implicit integration scheme for partial differential equations, where the source term for the time integration is equally weighted by the current and the projected values of the field. It is often unconditionally stable and has the best precision.


Because any arbitrary electric and magnetic fields are continuous quantities it is impossible to describe them numerically. Although methods are differerent the amount of information is reduced to an finite set of values, either the values at grid points, the amplitudes of orthonormal modes etc.

Eigenmode Solutions

The discretized Helmholtz equation (the Maxwell equation in frequency domain) has the format of an eigenvalue equations. Field solver are finding the different eigenvalues (aka frequency and damping constants) while the corresponding eigenvectors describe the unique field distribution of the eigenmodes.

Euler Method

The most simplest integration scheme for ordinary differential equations, where the solution is an extrapolation of the current value. Also often called forward or fully explicit integration scheme. It is not recommended due to its inherent instability and poor precision.

Explicit Integration

Integration scheme for partial differential equations, where the solution of the discretized field equation solely depends on the field values of the current time-step. Although the easiest and fastest method to implement, explicit integration scheme are often unconditionally unstable.

Finite-Difference Method

Discretization of the field equation on a regular grid. Any differential operator is replaced by its corresponding difference operator, resulting in an expression of close neighbor grid points. The operator can be represented as a matrix.

Finite-Element Method

The space is discretized by simple convex objects, over which the field is interpolated, e.g. by a cubic spline. The required continuity between neighbor elements and the applied differential operator results in a matrix equation for the interpolation constants.

Finite-Mode Method

The field is represented in a set of orthonormal functions (modes), suiting best the problem (e.g. waveguide modes or Gauss-Hermite modes). The field equations are reduced to decoupled integrals for the amplitudes of each mode.

Gauss-Seidel Relaxation

A relaxation scheme where the already calculated elements of the iterations are immediately used to calculated the remaining ones, and thus replacing the older ones.

Implicit Integration

Integration scheme for partial differential equations, where the solution of the discretized field equations has to agree with the assumption on the final field, which is used as the source term for the field equation. This requires typically a non-trivial inversion of the matrix, representing the spatial differential operator.

Jacobi Relaxation

A relaxation scheme where all elements of the next iteration are solely derived from the value of the previous iteration.

Multigrid Method

Relaxation method, where the poor convergence of low frequency components are solved, by projecting them to coarser grids, where they become high frequency components.

Paraxial Approximation

Approximation in the Maxwell equation, where a dominant oscillating term is spilt off the field. What remains is mainly the amplitude and phase of the field, which are slow-varying quantities and, thus, allowing larger integration step sizes.

Particle-in-Cell Simulation

Simulation, where the electric and magnetic fields are discretized on a regular grid, while the macro particles can freely propagated within. For evaluating the equation of motions either the field at the nearest grid point or a interpolation scheme is used.

Predictor-Corrector Method

An integration scheme for ordinary differential equations, where an initial guess for the solution is derived from the history of the particles. This guess is checked for consistency with the differential equation and, if needed, a correction is applied.


The process of iteration to find a solution to a matrix equations. Typically this method is used, when a direct solution (matrix inversion) is not possible due to memory limitation or singularities.

Richardson Interpolation Scheme

A advanced scheme for solving ordinary differential equations. Using Euler methods with very large step sizes and then reducing the step size, the convergence of the obtained solution is analysed with respect to the decreasing step size. This results in an automatic adjustment in the stepsize, depending on the curvature of the solution.

Runge-Kutta Integration

Higher-order integration scheme for ordinary differential equations at a given order. Typically the 4th order method is used, while the first order is identical to the Euler's Method with its poor accuracy.

Simplectic Integrator

A special integration scheme, which guarantees that certain conditions are fulfilled or conserved, e.g. the conservation of the particle energy. Typically it requires that only a reduced set of equations of motions are solved independently, while the solution for the remaining are derived from the applied constraints (e.g. energy conservation)


Preferred method to invert a matrix. Instead of the direct inversion the matrix is transformed into a product of three matrices: a unitary matrix, a diagonal matrix and another unitary matrix. The inversion is done by transposing the unitary matrices and inverting the diagonal elements of the central matrix.

Start-2-End Simulation

A chain of simulations, using different codes, modelling the evolution of an electron beam along a linear accelerator. The necessity arise due to the different type of physics along the beam line (Non-relativistic beam in a gun, coherent synchrotron radiation in a magnetic chicane, FEL amplification in an undulator), which can only be covered by highly specialized codes. The interface between the codes is the particle distribution itself or moments of the distribution, such as mean energy, energy spread, emittance etc.

Succesive Overrelaxation

Because relaxation schemes tends to converged to the solutions from one direction, the convergence can be enhanced by applying a weighting factor larger than one to the correction obtained from iteration.

Transport Matrix Method

In linear beam optics the particle position and momentum can be tracked through the beamline by applying matrix multiplication to the vector, containing the position of the particle in 6D phase space. The methods is simplectic, guaranteeing the conservation of the beam energy.