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Extra resources for Selected Topics in Particle Accelerators Vol V
Example text
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.
Selected Topics in Particle Accelerators Vol V
by Steven
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