A low-regularity Fourier integrator for the Davey-Stewartson II system with almost mass conservation

Abstract

In this work, we propose a low-regularity Fourier integrator with almost mass conservation to solve the Davey-Stewartson II system (hyperbolic-elliptic case). Arbitrary order mass convergence could be achieved by the suitable addition of correction terms, while keeping the first order accuracy in H^γ × H^γ+1 for initial data in H^γ+1 × H^γ+1 with γ > 1. The main theorem is that, up to some fixed time T, there exist constants τ₀ and C depending only on T and $||u|{|_{{L^\infty }\left( {(0,T);{H^{\gamma + 1}}} \right)}}$ such that, for any 0 < τ ≤ τ₀, we have that

$$||u({t_n}, \cdot ) - {u^n}|{|_{{H^\gamma }}} \le C\tau ,\,\,\,\,||v({t_n}, \cdot ) - {v^n}|{|_{{H^{\gamma + 1}}}} \le C\tau ,$$

where uⁿ and vⁿ denote the numerical solutions at t_n = nτ. Moreover, the mass of the numerical solution M(uⁿ) satisfies that

$$|M({u^n}) - M({u_0})| \le C{\tau ^5}.$$