In this paper, a scheme is presented for approximating solutions of non-linear degenerate parabolic equations which may contain discontinuous solutions. In the one-dimensional case, following the idea of the local discontinuous Galerkin method, first the degenerate parabolic equation is considered asa nonlinear system of first order equations, and then this system is solved using a fifth-order finite difference weighted essentially nonoscillatory (WENO) method for conservation laws. This is the first time that the minmod-limiter combined with weighted essentially nonoscillatory procedure has been applied to the degenerate parabolic equations. Also, it is necessary to mention that the new scheme has fifth-order accuracy in smooth regions and second-order accuracy near singularities. The accuracy, robustness, and high-resolution properties of the new scheme are demonstrated in a variety of multidimen- sional problems.

A double-GPU code is developed to accelerate WENO Schemes. The test problem is a compressible viscous flow. The convective terms are discretized using third-to ninth-order WENO Schemes and the viscous terms are discretized by the standard fourth-order central scheme. The code written in CUDA programming language is developed by modifying a single-GPU code. The OpenMP library is used for parallel execution of the code on both the GPUs. Data transfer between GPUs which is the main issue in developing the code, is carried out by defining halo points for numerical grids and by using a CUDA built-in function. The code is executed on a PC equipped with two heterogeneous GPUs. The computational times of different Schemes are obtained and the speedups with respect to the single-GPU code are reported for different number of grid points. Furthermore, the developed code is analyzed by CUDA profiling tools. The analyze helps to further increase the code performance.

This paper provides a fourth-order scheme for approximating solutions of non-linear degenerate parabolic equations that their solutions may contain discontinuity. In the reconstruction step, a fourth-order weighted essentially non-oscillatory (WENO) reconstruction in Legendre basis, written as a convex combination of interpolants based on different stencils, is constructed. In the one-dimensional case, the new fourthorder reconstruction is based on a four-point stencil. The most important subject is that one of these interpolation polynomials is taken as a quadratic polynomial, and the linear weights of the symmetric and convex combination are set as to get fourth-order accuracy in smooth areas. Following the methodology of the traditional WENO-Z reconstruction, the non-oscillatory weights is calculated by the linear weights. The accuracy, robustness, and high-resolution properties of the new procedure are shown by extensive numerical examples.

In this paper, a new fourth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed for the fractional differential equations which may contain non-smooth solutions at a later time, even if the initial solution is smooth enough. A set of Z-type non-linear weights is constructed based on the $L_1$ norm, yielding improved WENO scheme with more accurate resolution. The Caputo fractional derivative of order $\alpha$ is split into a weakly singular integral and a classical second derivative. The classical Gauss-Jacobi quadrature is employed for solving the weakly singular integral. Also, a new WENO-type reconstruction methodology for approximating the second derivative is developed. Some benchmark examples are prepared to illustrate the efficiency, robustness, and good performance of this new finite difference WENO-Z scheme.

In this article, we introduce a new method which allows utilizing all the available sub-stencils of a WENO scheme to increase the accuracy of the numerical solution of conservation laws while preserving the non-oscillatory property of the scheme. In this method, near a discontinuity, if there is a smooth sub-stencil with higher-order of accuracy, it is used in the reconstruction procedure. Furthermore, in smooth regions, all the sub-stencils of the same order of accuracy form the stencil with the highest order of accuracy as the conventional WENO scheme. The presented method is assessed using several test cases of the linear wave equation and one- and two-dimensional Euler’s equations of gas dynamics. In addition to the original weights of WENO Schemes, the WENO-Z approach is used. The results show that the new method increases the accuracy of the results while properly maintaining the ENO property

Weighted essentially non-oscillatory Schemes are among the most successful methods in numerical solutions to problems involving discontinuities. Since the accuracy of these Schemes mostly depends on their weights, various methods have been proposed to improve the weights. Although some numerical experiments show that the introduced improvements have some drawbacks, there is no suitable criterion to show which of them is superior to the others. In this study, we introduce a new way of assessing the performance of weighted essentially non-oscillatory Schemes: the Schemes’ performance in the long-time integration. This assessment can show the endurance of the scheme in preserving its maximum accuracy, which cannot be identified in a short time. Several methods from the literature are considered and is tested for the fifth, seventh, and ninth-order Schemes. First, the third-and fourth-order Runge-Kutta Schemes are used for the time integration. The results show the third-and fourth-order Runge-Kutta Schemes have a very small effect on the results even for long-time integration. In contrast, increasing the order of the spatial accuracy has a significant effect on the accuracy of the results. Furthermore, it can be observed that the parameters that have negligible effects on the results in the short time have considerable effects on the accuracy of the results in the long time, and choosing a proper value for them is crucial to obtain reasonable accurate results.

Weighted essentially non-oscillatory Schemes are among the most successful methods in numerical solutions to problems involving discontinuities. Since the accuracy of these Schemes mostly depends on their weights, various methods have been proposed to improve the weights. Although some numerical experiments show that the introduced improvements have some drawbacks, there is no suitable criterion to show which of them is superior to the others. In this study, we introduce a new way of assessing the performance of weighted essentially non-oscillatory Schemes: the Schemes’ performance in the long-time integration. This assessment can show the endurance of the scheme in preserving its maximum accuracy, which cannot be identified in a short time. Several methods from the literature are considered and is tested for the fifth, seventh, and ninth-order Schemes. First, the third-and fourth-order Runge-Kutta Schemes are used for the time integration. The results show the third-and fourth-order Runge-Kutta Schemes have a very small effect on the results even for long-time integration. In contrast, increasing the order of the spatial accuracy has a significant effect on the accuracy of the results. Furthermore, it can be observed that the parameters that have negligible effects on the results in the short time have considerable effects on the accuracy of the results in the long time, and choosing a proper value for them is crucial to obtain reasonable accurate results.

Two new higher order version of WENO Schemes are introduces and problems are solved to investigate problems containing shocks and disturbances in compressible flow. The solver is capable of solving conservation laws using WENO scheme of up to 7th order. The scheme is a recently developed version of the WENO-η-z method with a modified Global Smoothness Indicator (GSI) of 12th order of accuracy, aimed to decrease numerical dissipation over critical points. The code is primarily investigated trough solving several 1D and 2D problems, including the Sod’s shock tub, Lax’s problem, the Shu-Osher problem, which some are presented here as verification. The 2-D shock-bobble interaction and Richtmyer-Meshkov instability are solved as problems including shocks and disturbances, in which proposed methods are compared with both original WENO- η-z and two similarly modified methodes from recent literature. In these problems, the introduced scheme shows lower dissipation in comparison with the original versions, while having more acceptable stability and symmetry against other modifien versions.