Optimal convergence of a second-order low-regularity integrator for the KdV equation
Yifei Wu, Xiaofei Zhao
Abstract
Abstract In this paper, we establish the optimal convergence for a second-order exponential-type integrator from Hofmanová & Schratz (2017, An exponential-type integrator for the KdV equation. Numer. Math., 136, 1117–1137) for solving the Korteweg–de Vries equation with rough initial data. The scheme is explicit and efficient to implement. By rigorous error analysis, we show that the scheme provides second-order accuracy in $H^\gamma $ for initial data in $H^{\gamma +4}$ for any $\gamma \geq 0$, where the regularity requirement is lower than for classical methods. The result is confirmed by numerical experiments, and comparisons are made with the Strang splitting scheme.
Topics & Concepts
MathematicsIntegratorKorteweg–de Vries equationConvergence (economics)Scheme (mathematics)Type (biology)Exponential functionOrder (exchange)Applied mathematicsMathematical analysisNonlinear systemComputer scienceEconomicsEconomic growthPhysicsQuantum mechanicsBandwidth (computing)EcologyBiologyFinanceComputer networkNumerical methods for differential equationsAdvanced Numerical Methods in Computational MathematicsAdvanced Mathematical Physics Problems