Abstract
We consider a class of discrete operators introduced by O. Chodosh, acting on infinite sequences and mimicking standard properties of pseudo-differential operators. By using a new approach, we extend this class to finite or periodic sequences, allowing a general representation of discrete pseudo-differential operators obtained by finite differences approximations and easily transferred to time discretizations. In particular we can define the notion of order and regularity, and we recover the fundamental property, well known in pseudo-differential calculus, that the commutator of two discrete operators gains one order of regularity. As examples of practical applications, we revisit standard error estimates for the convergence of splitting methods, obtaining in some Hamiltonian cases no loss of derivative in the error estimates, in particular for discretizations of general waves and/or water-waves equations. Moreover, we give an example of preconditioner constructions inspired by normal form analysis to deal with the similar question for more general cases.
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Notes
Here \(\langle x\rangle ^2=1+|x|^2\).
Using typically the fact that the unbounded part can be diagonalized explicitly in Fourier to define mild-solutions, an example is given below.
Working with arbitrary integers is of course possible, by changing the structure of \(G_K\) according to the parity of K, see [12]
It is also easy to prove the stability estimates (4.3)
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Acknowledgements
During the preparation of this work the two authors benefited from the support of the Centre Henri Lebesgue ANR-11-LABX- 0020-01 and B.G. was supported by ANR -15-CE40-0001-02 “BEKAM” of the Agence Nationale de la Recherche. Furthermore B.G. thanks INRIA and particularly the MINGuS project for hosting him for a semester.
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A Young Inequality for Convolution
A Young Inequality for Convolution
Let \(x = (x_n)_{n \in \mathbb {Z}^d}\) and \(y = (y_n)_{n \in \mathbb {Z}^d}\) two sequences. We define \(z = x * y = (z_n)_{n \in \mathbb {Z}^d}\) the sequence
We also define for \(p \ge 1\),
And we recall the following Hölder inequality: for two sequences x, y
which is itself a consequence of the Young inequality for product: \(\forall a,b\ge 0\) we have \(ab \le \frac{a^p}{p} + \frac{b^q}{q}\). This Hölder inequality is easily generalized by induction to
for any sequences \(x^{(i)}\), \(i = 1, \ldots , N\) with \(N \in \mathbb {N}\).
Lemma A.1
For two sequences x and y indexed by \(\mathbb {Z}^d\), we have
Proof
Let us denote \(z = x * y\), we have
where we used the generalized trilinear Hölder inequality for the decomposition
Then we obtain
\(\square \)
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Faou, E., Grébert, B. Discrete Pseudo-differential Operators and Applications to Numerical Schemes. Found Comput Math (2024). https://doi.org/10.1007/s10208-024-09645-y
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DOI: https://doi.org/10.1007/s10208-024-09645-y
Keywords
- Pseudo-differential operators
- Numerical methods for discrete and fast Fourier transforms
- Finite difference methods
- Splitting methods.