Resonance-based schemes for dispersive equations via decorated trees

Abstract We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve nonlinear oscillations of partial differential equation (PDE) and approximate with high-order accuracy large class under lower regularity assumptions than classical techniques require. The key idea control frequency interactions in system up arbitrary high order thereby lies tailored decorated tree formalism. Our algebraic structures are close ones developed singular stochastic PDEs (SPDEs) structures. adapt them context by using novel decorations which encode dominant frequencies. proposed this article is new gives variant Butcher–Connes–Kreimer Hopf algebra on trees. observe similar Birkhoff type factorisation as SPDEs perturbative quantum field theory. allows single out optimise local error mapping it particular solution. use seems comparison literature. took advantage methods renormalisation theory extending via adjunction Taylor expansions. Now, through work, analysis taking these extended provides perspective them.