Exponential approximation space reconstruction weighted essentially nonoscillatory scheme for dispersive partial differential equations

In this work, we construct a fifth-order weighted essentially non-oscillatory (WENO) scheme with exponential approximation space for solving dispersive equations. A conservative third-order derivative formulation is developed directly using WENO spatial reconstruction procedure, and third-order TVD Runge-Kutta scheme is used for the evaluation of time derivative. This exponential approximation space consists a tension parameter that may be optimized to fit the specific feature of the characteristic data, yielding better results without spurious oscillations compared to the polynomial approximation space. A detailed formulation is presented for the construction of conservative flux approximation, smoothness indicators, nonlinear weights, and verified that the proposed scheme provides the required fifth convergence order. One- and two-dimensional numerical examples are presented to support the theoretical claims.