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This certainly results in many iterations that are necessary to calculate the propagation over time. Another way to overcome instabilities is to employ implicit integration methods, such as the Crank-Nicolson integration method [36, 182, 183]. 2) into the equation being integrated, the value to be determined occurs on both sides of the equation. 2) in space, and a trapezoidal rule in time; thus, can be regarded as an implicit method. 21 Chapter 2. 2) can be converted into a finite-dimensional system description in linear form xk+1 = Ak xk + Bk (ˆ uk + wxk ) .

2). However, throughout the entire work, we consider a certain space-time continuous system occuring in many applications, the convection-diffusion system. 1) convection term where r := [x, y]T ∈ R2 denotes the spatial coördinate and p(r, t) and s(r, t) are the space-time continuous system state and the space-time continuous system input. The vector v := [vx , vy ]T ∈ R2 represents the homogeneous convection field. The diffusion coefficient α ∈ R is characterized by specific material properties, such as the medium density ρ, the heat capacity cp , and the thermal conductivity k, according to α := κ/(ρ cp ).

Reconstruction and Interpolation of Space-Time Continuous Systems distributedparameter (a) Conversion of system description (b) Shape functions (Stochastic) partial diff. equation ∂p(r, t) − α∇2 p(r, t) = γ s(r, t) ∂t Piecewise linear y functions x discretetime lumpedparameter Orthogonal polynomials (nodal expansion) Shape functions y x (Stochastic) ordinary diff. 8: (a) Visualization of the individual stages for the conversion of stochastic partial differential equations into a time-discrete system model in state-space form.

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Selected Topics in Particle Accelerators Vol V

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