Search

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. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we have proposed a new problem independent discontinuity locator for hybridization. Along with this, a new global smoothness indicator is proposed. The hybrid weighted essentially nonoscillatory (WENO) scheme works on the principle of applying linear upwind scheme in smooth regions and WENO algorithm in nonsmooth regions to save computational cost, which is reflected in the time taken for the computation of the numerical solution. A problem independent discontinuity locator has been designed based on the properties of the existing smoothness indicators. In addition, to attain the required order of accuracy, a new global smoothness indicator is proposed, which produces a solution with better resolution at discontinuities. Numerical experiments have been included to demonstrate the application of the proposed theory to some benchmark problems. It has been shown that the newly proposed hybrid WENO scheme works faster and gives a solution of higher resolution than WENO‐JS, WENO‐Z, and few other more recent third‐order WENO schemes. [ABSTRACT FROM AUTHOR]

Summary: A new third‐order WENO scheme is proposed to achieve the desired order of convergence at the critical points for scalar hyperbolic equations. A new reference smoothness indicator is introduced, which satisfies the sufficient condition on the weights for the third‐order convergence. Following the truncation error analysis, we have shown that the proposed scheme achieves the desired order accurate for smooth solutions with arbitrary number of vanishing derivatives if the parameter ε satisfies certain conditions. We have made a comparative study of the proposed scheme with the existing schemes such as WENO‐JS, WENO‐Z, and WENO‐N3 through different numerical examples. The result shows that the proposed scheme (WENO‐MN3) achieves better performance than these schemes. [ABSTRACT FROM AUTHOR]