Linearize or Discretize First? Error Estimates under Mesh-Uniform Regularity and Exponential Stability for the KdV Equation with Boundary Dissipation

Abstract

We study the Korteweg-de Vries equation on a bounded interval with boundary dissipation, and its spatial semidiscretization by a weighted cubic Hermite Galerkin method. A small nonlinear wave can be approximated by a linear discrete simulation in two orders: linearize first and then discretize, or discretize first and then linearize. Both orders give the same total error, but they divide it differently between linearization and discretization. The difference of the two divisions is the ordering error. We show that it is quadratic in the size of the initial datum and carries a positive power of the mesh size, for all time. For the continuous problem, the solution map is analytic, with a power series that converges for all time in exponentially weighted trajectory norms. The linearized semigroup has Crandall-Pazy regularity with the exact index 3/2. For the weighted scheme, exponential stability, Kato smoothing, Crandall-Pazy regularity and analyticity of the solution maps hold uniformly in the mesh size. These properties give error estimates in which the error and the datum are measured in the same norm: in the energy norm, in $H_0^1$ and in $H^2$. In exponentially weighted time-integrated norms the semidiscrete solutions converge uniformly on balls, at the rate $h^{1/6}$ in $L^4(0,\infty;L^2)$ for data in $L^2$ and at the rate $h^{2/3}$ in $L^2(0,\infty;H^1)$ for data in $H_0^1$. On every interval $[t_0,\infty)$, $t_0>0$, same-norm convergence holds uniformly on balls at algebraic rates. The ordering error is of order $h^{1/6}$ times the square of the energy norm of the datum, uniformly for all $t\ge0$. For the unweighted scheme, a grid wave packet of zero group velocity destroys mesh-uniform Kato smoothing and Crandall-Pazy regularity.