The standard technique is to enforce the boundary conditions exactly, i.e. up to machine precision. Let us suppose here that the time-dependent PDE (116), which we want to solve, is well posed with boundary condition

where is a given function. We give here some examples, with the forward Euler scheme (118) for time discretization.In the collocation method, the values of the approximate solution at (Gauss–Lobatto type) collocation points are determined by a system of equations:

where the value at the boundary is directly set to be the boundary condition.In the tau method, the vector is composed of the coefficients at the -th timestep. If we denote by the -th coefficient of applied to , then the vector of coefficients is advanced in time through the system:

the last equality ensures the boundary condition in the coefficient space.

As shown in the previous examples, the standard technique consists of neglecting the solution to the PDE for one degree of freedom, in configuration or coefficient space, and using this degree of freedom in order to impose the boundary condition. However, it is interesting to try and impose a linear combination of both the PDE and the boundary condition on this last degree of freedom, as is shown by the next simple example. We consider the simple (time-independent) integration over the interval :

where is the unknown function. Using a standard Chebyshev relation (161)collocation method (see Section 2.5.3), we look for an approximate solution as a polynomial of degree verifyingWe now adopt another procedure that takes into account the differential equation at the boundary as well as the boundary condition, with verifying (remember that ):

where is a constant; one notices when taking the limit , that both systems become equivalent. The discrepancy between the numerical and analytical solutions is displayed in Figure 23, as a function of that parameter , when using . It is clear from that figure that there exists a finite value of () for which the error is minimal and, in particular, lower than the error obtained by the standard technique. Numerical evidence indicates that . This is a simple example of weakly imposed boundary conditions, with a penalty term added to the system. The idea of imposing boundary conditions up to the order of the numerical scheme was first proposed by Funaro and Gottlieb [85] and can be efficiently used for time-dependent problems, as illustrated by the following example. For a more detailed description, we refer the interested reader to the review article by Hesthaven [115].Let us consider the linear advection equation

where is a given function. We look for a Legendre collocation method to obtain a solution, and define the polynomial , which vanishes on the Legendre–Gauss–Lobatto grid points, except at the boundary :
Living Rev. Relativity 12, (2009), 1
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