This talk focuses on the construction and analysis of numerical schemes for computing rough solutions of nonlinear dispersive equations, such as the Korteweg–De Vries (KdV) equation and the nonlinear Schrödinger (NLS) equation, with regularity below H¹. To establish stability estimates under low-regularity conditions, we introduce continuous formulations of the numerical schemes, wherein the discrete numerical solution is recast as the solution of a perturbed continuous equation. This approach reduces the discrete stability analysis to analyzing the stability of the continuous equation with respect to perturbations. Consequently, stability can be established by utilizing continuous-level Bourgain or Strichartz estimates, thereby circumventing the restrictive CFL conditions typically required when studying the stability of numerical schemes at the discrete level. Furthermore, a high- and low-frequency splitting technique may be further employed to improve the convergence rates of the numerical solutions. For the KdV equation, we prove that the proposed method converges in L² with order β−ϵ under the regularity condition u∈C([0,T];Hᵝ) for β∈(0,1]. Similarly, for the NLS equation, the method converges in L² with order 1.5β−ϵ under the regularity condition u∈C([0,T];Hᵝ) for β∈(0,1/2].
