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In this paper, we present a weighted essentially non-oscillatory (WENO) scheme for dispersive equations which may generate physical high-frequency oscillation in the non-smooth interface. The third derivative term is approximated directly by a conservative flux difference. A finite-difference WENO scheme of fifth-order is constructed for the discretization of spatial differentiation. The wave behavior of linear and nonlinear dispersion equations is simulated by using the proposed scheme in space direction and the third-order TVD Runge–Kutta method in the time direction. Numerical examples demonstrate the accuracy and good performance of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new high-order finite volume method for solving the multicomponent fluids problem with Mie–Grüneisen EOS. Firstly, based on the cell averages of conservative variables, we develop a procedure to reconstruct the cell averages of the primitive variables in a high-order manner. Secondly, the high-order reconstructions employed in computing numerical fluxes are implemented in a characteristic-wise manner to reduce numerical oscillations as much as possible and obtain high-resolution results. Thirdly, advection equation within the governing system is rewritten in a conservative form with a source term to enhance the scheme's performance. We utilize integration by parts and high-order numerical integration techniques to handle the source terms. Finally, all variables are evolved by using Runge–Kutta time discretization. All steps are carefully designed to maintain the equilibrium of pressure and velocity for the interface-only problem, which is crucial in designing a high-resolution scheme and adapting to more complex multicomponent problems. We have performed extensive numerical tests for both one- and two-dimensional problems to verify our scheme's high resolution and accuracy. • A new uniform high-order FVM for the multicomponent fluids with Mie–Grüneisen EOS. • High-order reconstruction for the cell averages of the primitive variables. • High-order numerical integration techniques is applied to handle the source terms • Maintain the equilibrium of pressure and velocity for the interface-only problem. [ABSTRACT FROM AUTHOR]

Abstract: A class of fifth-order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one dimensional non-linear hyperbolic conservation law systems, was developed and applied as limiters for the Runge–Kutta discontinuous Galerkin (RKDG) methods in [J. Comput. Phys. 193 (2003) 115]. In this paper, we extend the method to solve two dimensional non-linear hyperbolic conservation law systems. The emphasis is again on the application of such HWENO finite volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional Burgers’ equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods. [Copyright &y& Elsevier]

• Construction of a new type of high order HWENO methods on unstructured meshes. • More compact stencil than regular WENO schemes. • Arbitrary positive linear weights and single WENO polynomial on each triangle. • Numerical tests demonstrating the good performance of the scheme. In this paper, a new type of high order Hermite weighted essentially non-oscillatory (HWENO) methods is proposed to solve the Hamilton–Jacobi (HJ) equations on unstructured meshes. We use a fourth order accurate scheme to demonstrate our procedure. Both the solution and its spatial derivatives are evolved in time. Our schemes have three advantages. First, they are more compact than the one in [38] as more information is used at each node which allows us to achieve the same high order accuracy with a more compact stencil. Second, the new HWENO approximation on the unstructured mesh allows arbitrary positive linear weights, which enhances the stability of our scheme. Third, the new HWENO procedure produces an approximation polynomial on each triangle, which allows us to compute all the spatial derivatives at the three nodes of each triangle based on this single polynomial, instead of computing each derivative individually with different linear weights in the classical HWENO framework, which improves the efficiency of our scheme. Extensive numerical experiments are performed to verify the accuracy, high resolution and efficiency of this new scheme. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we develop the Runge–Kutta discontinuous Galerkin (RKDG) methods with the finite volume weighted essentially non-oscillatory (WENO) reconstruction as limiters to solve for the unsteady cavitating flow under the employment of the isentropic one-fluid model. To treat the cavitating flow and suppress the possible spurious oscillations in the vicinity of the cavitation boundary, the TVB limiter is used as an indicator to detect the “troubled cells” and hence take the advantage of utilizing the WENO reconstruction for the freedoms of the RKDG methods. Numerical results are provided to illustrate the viability of these procedures. [Copyright &y& Elsevier]

• A new fifth order hybrid WENO scheme is presented. • New indicator to switch using WENO or linear approximation. • No characteristic projection for linear approximation. • More efficient and smaller numerical errors than the simple WENO. In [28], a simple fifth order weighted essentially non-oscillatory (WENO) scheme was presented in the finite difference framework for the hyperbolic conservation laws, in which the reconstruction of fluxes is a convex combination of a fourth degree polynomial with two linear polynomials. In this follow-up paper, we propose a new fifth order hybrid weighted essentially non-oscillatory (WENO) scheme based on the simple WENO. The main idea of the hybrid WENO scheme is that if all extreme points of the reconstruction polynomial for numerical flux in the big stencil are located outside of the big stencil, then we reconstruct the numerical flux by upwind linear approximation directly, otherwise use the simple WENO procedure. Compared with the simple WENO, the major advantage is its higher efficiency with less numerical errors in smooth regions and less computational costs. Likewise, the hybrid WENO scheme still keeps the simplicity and robustness of the simple WENO scheme. Extensive numerical results for both one and two dimensional equations are performed to verify these good performance of the proposed scheme. [ABSTRACT FROM AUTHOR]