2024/12/22 by Biswas, Abhijit, Ketcheson, David I., Ranocha, Hendrik +1 · 3 citations
#65L04 #65L20 #65M06 #65M12 #65M22 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2412.17117
We study the recently-proposed hyperbolic approximation of the Korteweg-de Vries equation (KdV). We show that this approximation, which we call KdVH, possesses a rich variety of solutions, including solitary wave solutions that approximate KdV solitons, as well as other solitary and periodic solutions that are related to higher-order water wave models, and may include singularities. We analyze a class of implicit-explicit Runge-Kutta time discretizations for KdVH that are asymptotic preserving, energy conserving, and can be applied to other hyperbolized systems. We also develop structure-preserving spatial discretizations based on summation-by-parts operators in space including finite difference, discontinuous Galerkin, and Fourier methods. We use the entropy relaxation approach to make the fully discrete schemes energy-preserving. Numerical experiments demonstrate the effectiveness of these discretizations.