As an algorithm, the tau-method is a synonym for expanding the residual function as a series of Chebyshev polynomials and then applying the boundary conditions as side constraints. Thus it’s indistinguishable from what we have earlier described as the Galerkin’s method with “boundary bordering”. - Chebyshev and Fourier Spectral Methods, John P. Boyd
A mean-weighted residual method (MWR) is a tau-method if the “test functions” are Chebyshev polynomials and the inner product is the usual Chebyshev integral inner product with weight function .
For example, to solve a second order ODE with , the -method would impose the following mean-weighted residual constraints
and boundary constraints
where is the -th Chebyshev polynomial of the first kind, and is the term Chebyshev series approximation to the solution .
Lanczos recognized that there are two ways of attacking a differential equation. The obvious approach is to compute an approximation solution to the exact, unmodified differential equation. The second is to compute the exact solution to a modified of the original differential equation. If the “modification” is small, then the solution to the modified problem will be a good approximation to that of the original problem. The second strategy — to solve approximate equations exactly — is the philosophy of the -method.
Tau-Approximation for a Rational Function
To approximate a given rational function by a polynomial of degree , where and are polynomials of degree and respectively, a natural idea is to minimize the linearized error
Since is a polynomial of degree and is a polynomial of degree , error can not be zero for all in general: we need more conditions on coefficients of to garantee than the number of unknown coefficients in .
Lanczos observed that there is an exact, polynomial solution to the perturbed problem
where is a polynomial of degree with undetermined coefficients. The above equation hence has polynomials of degree on both sides with a total of unknowns. Matching the powers of on both sides of this equation gives linear equations to determine the unknown coefficients of and .
The secret to success is to choose — more accurately, to choose of the coefficients of — such that the perturbation is small in some appropriate sense. The most natural choice is to set all for all . In this case, is the usual power series approximation to . The trouble with this choice is that is highly non-uniform, because it’s for small where this “power series perturbation” is very small near the origin, but grows rapidly for large . Hence this native choice is not a good one for approximating in general.
Lanczos’ second key observation is that the Chebyshev polynomials oscillate as uniformly as possible on their canonical interval , which makes them the best choice to be the basis functions of in the sense of uniform approximation. It follows that if we define
the perturbation will spatially uniform in magnitude on and so will the error .
If we write
then the are determined by the linear equations
and thus for .
Remark. There are several important points:
It’s not necessary to explicitly compute the -coefficients in order to determine the approximation .
The -approximation is not simply the Chebyshev expansion of .
Equation (Lanczos Perturbation) is not the way Lanczos originally performed the calculation.
The -coefficients are useful for a posteriori error analysis because
Tau-Method for Differential Equations
In this section (mainly comes from [3]), we consider solve the following polynomial coefficients linear differential equation
The Lanczos’ procedure, instead of looking for an th order approximate solution of by truncating an infinite power series expansion, tries to find an exact polynomial solution to a modified version of the equation, called the perturbed equation, which is obtained by adding to the right-hand side of a polynomial perturbation term. This term is chosen in such a way that it becomes possible to find a power series solution with only finite number of terms different from zeros.
Lanczos introduced in [2] a sequence of canonical polynomials associated with , which are defined by means of the functional relation
As a perturbation term to , Lanczos takes a polynomial , where is, in some sense, an approximation to the right-hand side of by means of a polynomial of degree , and is a polynomial conveniently chosen in order to satisfy the supplementary conditions of the problem.
The solution of
in terms of the canonical polynomials associated with is given by
The choice of the coefficients depends on the approximation problem we are trying to solve. If we are interested in an approximation of the solution of in the large over the finite interval , we could take as the coefficients of the Chebyshev polynomial of order defined in as it gives the best uniform approximation of order to in .
Canonical Polynomial
Let be the class of linear differential operators with polynomial coefficients and finite order. Given any , define the generating polynomial of order assciated with as
Furthermore, we define the height of to be
and the numerical set
Let . As if and the coefficients of are polynomial functions of , the degree of which does not exceed the order of , and is an upper bound for .
Let be the set of finite linear combinations
and be the set of index such that there is no polynomial of degree in , called the set of undefined index. It follows that and is finite. The set is used to determine whether a given polynomial right-hand side residual can be expressed as the image of some polynomial under .
The subspace spanned by , is the subspace of residuals of , i.e., is the subspace generated by . Let be the subspace spanned by the polynomial solutions of equation , where is the order of .
Definition. is called a canonical polynomial of order associated with if
where , is the residual polynomial of . The index takes into account the fact that there may have multiple canonical polynomials.
According to the definition, multiple canonical polynomials with respect to some given differs by a polynomial solution of . Now let be the equivalent relation defined on such that if and only if , then we define the quotient set
as the sequence of Lanczos’ classes of equivalence associated with the operator . Obviously, we have if . Moreover, let , then the following theorem holds.
Theorem. There is a one-to-one correspondence between and , i.e., the bijection is given by
Equivalent Formulation
There is an equivalent formulation of the tau-method for differential equations, which is similar to the tau-approximation for rational functions. From now on, we consider the general differential equation
on some spatial domain with boundary , along with the boundary conditions on and initial condition . The operator is a linear (spatial, that is, ) differential operator and is a linear (time-independent, that is, ) boundary operator.
It’s assumed that, for each , exact solution of is an element of a Hilbert space with some inner product and norm . For each , the solution , as a function of , belongs to the subspace of , where
Do not require but only . The operator is typically an unbounded differential operator, whose domain is dense in, but smaller than .
The semi-discrete approximations to to be studied here are of the form
where belongs to an -dimensional subspace of , and is a linear operator from to of the form
Here is the projection operator from to . Assume if and and .
Similar to Galerkin method, the approximation by tau-method is sought in the form of the truncated series
where the basis functions are assumed to be elements of a complete set of orthonormal functions and is the number of independent boundary constraints that must be imposed. However, there is an important difference between the tau-method and the Galerkin method: the tau-method does not require the basis functions to satisfy the boundary conditions individually, instead it uses the “boundary bordering” strategy to impose the boundary conditions. The boundary constraints
are imposed as part of the conditions determining the expansion coefficients of a solution in .
As a MWR method, the tau-method requires that the residual function satisfy the following mean-weighted residual constraints
Since the basis functions are orthonormal, the above constraints can be written as
In conclusion, the tau-method for differential equations construct the approximation with coefficients by solving the differential equations above and imposing the boundary constraints.
In fact, the resulting approximation is the exact solution to the modified problem
which lies in for all . For each initial value problem and choice of orthonormal basis (and associated inner product), there is a (normally unique) choice of -coefficients such that , namely
for , the remaining -coefficients are determined by the boundary constraints
and the dynamical constraints above.
Reference
John P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd Edition. Dover Publications, 2001.
Lanczos C. Trigonometric interpolation of empirical and analytical functions. Journal of Mathematics and Physics, 1938, 17(1-4): 123-199.
Ortiz E. L. The tau method. SIAM Journal on Numerical Analysis, 1969, 6(3): 480-492.
Gottlieb D, Orszag S. A. Numerical analysis of spectral methods: theory and applications. SIAM, 1977.
