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In this paper, the authors introduce a new Weighted Essentially Nonoscillatory (WENO) scheme. This scheme is founded on exponential functions and utilizes arc-length smoothness indicators. The primary purpose of this WENO scheme is to provide accurate approximations for the viscosity numerical solutions of Hamilton–Jacobi equations. The arc-length smoothness indicators are derived from the derivatives of reconstructed polynomials within each sub-stencil. These smoothness indicators play a crucial role in approximating the viscosity numerical solutions of Hamilton–Jacobi equations, ensuring high-resolution results and minimizing absolute truncation errors. Numerous numerical tests have been carried out and presented to demonstrate the performance capabilities and numerical accuracy of the proposed scheme, comparing it to several traditional WENO schemes.

The recently developed physics-informed machine learning has made great progress for solving nonlinear partial differential equations (PDEs), however, it may fail to provide reasonable approximations to the PDEs with discontinuous solutions. In this paper, we focus on the discrete time physics-informed neural network (PINN), and propose a hybrid PINN (hPINN) scheme for the nonlinear PDEs. In this approach, the local solution structures are classified as smooth and nonsmooth scales by introducing a discontinuity indicator, and then the automatic differentiation technique is employed for resolving smooth scales, while an improved weighted essentially nonoscillatory (WENO) scheme is adopted to capture discontinuities. We then test the present approach by considering the viscous and inviscid Burgers equations, and it is shown that compared with original discrete time PINN, the present hPINN approach has a better performance in approximating the discontinuous solution even at a relatively larger time step.

In this study, a high-robustness hybrid scheme of weighted essentially non-oscillatory (WENO) scheme with a modified tangent of hyperbola for interface capturing (THINC) algorithm is developed for compressible multicomponent flow on general curvilinear grids. Numerical errors induced by mesh deformation bring loss of numerical accuracy and simulation instability, resulting in inaccurate results such as interface distortion, numerical oscillations or even simulation failure. To address this issue, the WENO scheme combined with an improved THINC strategy is designed to alleviate these errors and maintain the high resolution of interfaces. A modified THINC algorithm is developed for the non-uniform grids, in which the steepness parameter is scaled adaptively according to varying grid spacings. This approach is capable of reducing numerical dissipations for interface reconstruction. The overestimated quasi-conservative WENO formulation are employed to hold the equilibriums of velocity, pressure, and temperature at the material interface. Numerical validations are tested on non-uniform grids with various randomness amplitudes to verify the effectiveness in one- and two-dimensional benchmark problems showing the better performances in shock- and interface-capturing capabilities.

Traveling waves of a class of higher-order traffic flow models with viscosity are studied with the reduction perturbation method, which leads to the well-known Kortweg–de Vries equation and the approximate solitary wave solution to the model. The fifth-order accuracy weighted essentially nonoscillatory scheme is adopted for comparison between the analytical and numerical results. The numerical tests show that the solitary wave evolves with little deformation of its profile and that a globally perturbed equilibrium traffic state is able to evolve into a profile similar to that of a solitary wave, which is identified by the same total number of vehicles on the ring road. These results are compared with those in the literature and demonstrate that the approximation to the model is more accurate.

The prediction of heat transfer for blunt bodies in hypersonic flows remains a great challenge. In particular, the uncertainties are larger in the leeside due to the complexity of the wake flow. Generally, the heat transfer is over-predicted using the Reynolds-averaged Navier–Stokes (RANS) models. In this paper, the improved delayed detached eddy simulation (IDDES) method is used to simulate the Mach 6 flow around a scaled spherical capsule model. In addition, a low dissipative WENO scheme is used for inviscid fluxes and dual-time stepping method is applied for time advancement. Results are compared to experimental data for mean and instantaneous heat transfer in the leeside of the aftbody. It is shown that the integrated error is 75.49% for RANS while 35.69% for IDDES method. Moreover, the multi-scale structures in the separation region are also resolved well by the IDDES method.

A two-dimensional numerical model using the fifth-order weighted essentially nonoscillatory (WENO) scheme is presented in order to estimate topography change due to tsunami with high accuracy. In the model, the Cartesian coordinate system is adopted, and the fractional area/volume obstacle representation (FAVOR) method is introduced into the governing equations in consideration of applying the estimation to such as harbor shape with complex topography. In order to verify the validity and applicability of the model, it is applied to small-scale laboratory experiments and to large-scale actual topography change. Consequently, although the model cannot reproduce local scouring around breakwater where three-dimensional flow is developed, it is clarified that the model can reproduce the topography change well by contracted flow around a harbor.