# Resonance-based schemes for dispersive equations via decorated trees

@article{Bruned2022ResonancebasedSF, title={Resonance-based schemes for dispersive equations via decorated trees}, author={Yvain Bruned and Katharina Schratz}, journal={Forum of Mathematics, Pi}, year={2022}, volume={10} }

Abstract We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in…

## 26 Citations

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Low regularity integrators via decorated trees

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We introduce a general framework of low regularity integrators which allows us to approximate the time dynamics of a large class of equations, including parabolic and hyperbolic problems, as well as…

Approximations of dispersive PDEs in the presence of low-regularity randomness

- Mathematics, Computer ScienceArXiv
- 2022

A novel combinatorial structure called paired decorated forests which are two decorated trees whose decorations on the leaves come in pair are introduced which allows for low regularity approximations to the expectation E ( | u k ( τ, v η ) | 2 ) , where u k denotes the k -th Fourier coeﬃcient of the solution u of the dispersive equation and v x the associated random initial data.

## References

SHOWING 1-10 OF 79 REFERENCES

Numerical Dispersive Schemes for the Nonlinear Schrödinger Equation

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2009

The convergence of the two-grid method for nonlinearities that cannot be handled by energy arguments and which, even in the continuous case, require Strichartz estimates is proved.

Uniformly accurate numerical schemes for highly oscillatory Klein–Gordon and nonlinear Schrödinger equations

- Computer Science, MathematicsNumerische Mathematik
- 2015

A general strategy to construct numerical schemes which are uniformly accurate with respect to the oscillation frequency is presented, enabling to simulate the oscillatory problem without using any mesh or time step refinement and the orders of the authors' schemes are preserved uniformly in all regimes.

A Fourier Integrator for the Cubic Nonlinear Schrödinger Equation with Rough Initial Data

- MathematicsSIAM J. Numer. Anal.
- 2019

This work presents a new type of integrator that is based on the variation-of-constants formula and makes use of certain resonance based approximations in Fourier space that can be efficiently evaluated by fast Fourier methods.

A theory of regularity structures

- Mathematics
- 2014

We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each…

Renormalising SPDEs in regularity structures

- Mathematics
- 2017

The formalism recently introduced in arXiv:1610.08468 allows one to assign a regularity structure, as well as a corresponding "renormalisation group", to any subcritical system of semilinear…

Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity

- Mathematics, Computer ScienceFound. Comput. Math.
- 2021

A rigorous error analysis is performed and better convergence rates at low regularity are established than known for classical schemes in the literature so far for this new filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation.

Convergence Analysis of High-Order Time-Splitting Pseudospectral Methods for Nonlinear Schrödinger Equations

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2012

This work provides a stability and error analysis of high-accuracy discretizations that rely on spectral and splitting methods and relies on a general framework of abstract nonlinear evolution equations and fractional power spaces defined by the principal linear part.

Algebraic renormalisation of regularity structures

- MathematicsInventiones mathematicae
- 2018

We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a…

B-series and Order Conditions for Exponential Integrators

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2005

A general format of numerical ODE-solvers which include many of the recently proposed exponential integrators is introduced and a general order theory for these schemes is derived in terms of $B$-series and bicolored rooted trees.

Renormalisation of Stochastic Partial Differential Equations

- Mathematics
- 2020

We present the main ideas of the renormalisation of stochastic partial di ﬀ erential equations, as it appears in the theory of regularity structures. We informally discuss the regularisation of the…