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1.4 Numerical discretization


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Numerical solutions are obtained by evolving a discrete set of values $ \{f_j\}$ , $ j=1,N$ with small steps in time $ \Delta t$ to approximate what really should be a continuous function of space and time $ f(x,t)$ , $ x\in[a;b]$ , $ t\ge t_0$ . Unfortunately, there is no universal method. Rather than blindly adopting a local favorite, your choice really should depend on

  • the structure of the solution (continuity, regularity, precision), post-processing (filters) and the diagnostics (Fourier spectrum) that are expensive but might be required anyway,
  • the boundary conditions that can be difficult to implement in some methods,
  • the structure of the differential operator (the analytic formulation, the computational cost in memory$ \times$ time, the numerical stability) and the computer architecture (vectorization, parallelization).
This course is a lot about advantages and limitations from a variety of methods. By analyzing model problems in a simple 1D slab, you learn the tricks that are important to know before you work with a higher number of dimensions. The broader perspective allows you to choose an optimal solution, where you can exploit the advantages and work around the limitations for your particular problem.



Subsections SYLLABUS  Previous: 1.3.5 Schrödinger  Up: 1 INTRODUCTION  Next: 1.4.1 Convergence, Richardson extrapolation
      
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