We the algorithm this switching takes place.3) Computational

We investigate the advantages and disadvantages associated with the spectral and finite difference schemes, in terms of accuracy, computational efficiency, implementation and boundary conditions 12.1) Accuracy:• Finite Differences: determined by the discretization step ?x. Accuracy is easy to compute but generally muchworse than spectral methods.• Spectral Method: they rely on a global expansion and are often called spectrally accurate. In particular, spectral methods have infinite order accuracy, so they are generally of much higher accuracy than finite differences.2) Implementation:• Finite Differences: the greatest difficulty in implementing the finite difference schemes is generating the dis-cretization state-space. In linear systems, this means obtaining the sparse matrix. The non-linearities in thepresent case made the calculations much slower.• Spectral Method: the difficulty comes from the continual switching between time/space domain and the spectral domain, thus it is imperative to know exactly when and where in the algorithm this switching takes place.3) Computational time:• Finite Differences: The computational time for finite differences is determined by the size of the state-space when solving the ODEs. In the present case, due to the complexity of the system, the simulation was very time consuming.• Spectral Method: The FFT algorithm is an O(N log(N )) operation, so it is (almost) always guaranteed to be faster than the finite difference method which is O(N 2 ) 12. This is clearly reflected in time records of the twomethods,4) Boundary Conditions:• Finite Differences: implementing the generic boundary conditions ?u(L) + ? ?u(L) = ? is easily done in ?x the finite difference framework. Even more complicated computational schemes may be considered. Generally,any computational domain which can be constructed of rectangles is easily handled by finite difference methods 12. The results of this implementation are shown in Figures 15, 16.• Spectral Method: boundary conditions are the main limitation of this method. Specifically, only periodic boundary conditions can be considered, so the Tokamak model is not feasible in this framework (see Figure17, 18). An alternative is to use Chebychev Transform, a spectral method that allows the use of non-periodicboundaries.


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