Low-regularity integrators for the “good” Boussinesq equation with linearly decreasing additional derivative requirements

In this paper, we propose first- and second-order low-regularity exponential-type integrators (LREIs) for the “good” Boussinesq equation. Compared to existing exponential-type integrators, our proposed LREIs achieve the desired convergence rates under lower regularity conditions. These new integrators are developed using a second-order approximation of the integral in the Duhamel formula, rather than approximating the integrand, which is the approach taken by previous low-regularity exponential-type integrators. Through the application of certain technical inequalities, we demonstrate that our newly proposed integrators can attain the same accuracy under reduced regularity assumptions. We highlight two key properties of our schemes: (1) For the first-order integrator (LREI1), the additional regularity required to achieve an error of order [Formula: see text] in [Formula: see text] ([Formula: see text]) decreases linearly to zero as [Formula: see text] increases. Moreover, the convergence order in [Formula: see text] increases linearly from zero to one as [Formula: see text] transitions from [Formula: see text] to [Formula: see text], without any loss of regularity. (2) For the second-order integrator (LREI2), the extra regularity required to achieve an error of order [Formula: see text] in [Formula: see text] ([Formula: see text]) diminishes linearly to [Formula: see text] as [Formula: see text] increases. Numerical results are presented to support our theoretical findings, demonstrating an improvement in accuracy compared to previous methods, especially for solutions exhibiting lower regularity.