A high‐order weighted essentially nonoscillatory scheme based on exponential polynomials for nonlinear degenerate parabolic equations

Abstract

In this research the numerical solution of nonlinear degenerate parabolic equations is investigated by a new sixth-order finite difference weighted essentially nonoscillatory (WENO) based on exponential polynomials. In smooth regions, the new scheme, named as EPWENO6, can achieve the maximal approximation order while in critical points it does not lose its accuracy. In order to better approximation near steep gradients without spurious oscillations, the EPWENO6 scheme is designed by the exponential polynomials that are incorporated into the WENO reconstruction. In design of nonlinear weights that play a very important role in WENO reconstructions, a global smoothness indicator using generalized undivided differences is introduced. To analyze the convergence order of the EPWENO6 method in full detail, the Lagrange-type exponential functions have been used. By comparing the EPWENO6 scheme and other existing WENO schemes for solving degenerate parabolic equations, it can be seen that the new scheme offers better results while its computational cost is less. To illustrate the effectiveness of the proposed scheme, a number of numerical experiments are considered.

A sixth-order finite difference WENO scheme for non-linear degenerate parabolic equations: an alternative technique

“…To measure the smoothness of polynomials q r ( x ) with r = 0 ,1,2 in the interval [ x j , x j +1 ], smoothness indicators represented by β r ( r = 0 ,1,2) are used. They are defined and calculated as follows: (Liu et al , 2011; Abedian et al , 2013; Abedian, 2021a, 2021b; Abedian and Dehghan, 2022): …”

“…In this section, the second derivative in equation ( 1) is conservatively discretized with the WENO procedure as in Abedian et al (2013), Abedian (2021aAbedian ( , 2021b and Abedian and Dehghan (2022) but with a different idea, which we will discuss in detail below.…”

“…It has been observed that the approximation methods based on exponential polynomials are more suitable than those based on algebraic polynomials when interpolating data with rapid gradients (Ha et al , 2016). Therefore, Abedian and Dehghan proposed an improved sixth-order WENO procedure for solving degenerate parabolic equations by incorporating exponential polynomials (Abedian and Dehghan, 2022). In smooth regions, the developed method obtains the maximal approximation order without loss of accuracy, even at critical points.…”

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

“…This limitation can result in significant numerical dissipation when interpolating data with rapid gradients, which hinders the ability to produce sharp edges. To overcome this limitation, researchers [20][21][22][23][24] have explored the use of other types of basis functions, such as exponential polynomials, which have been shown to yield better results in terms of producing sharp edges and reducing numerical dissipation.…”

“…The comparative analysis of numerical outcomes between methodologies based upon exponential/trigonometric/ algebraic polynomial constructions for hyperbolic conservation laws [20][21][22][23] and nonlinear degenerate parabolic equations [24][25][26] has been studied in literature. In instances involving interpolation of data manifesting rapid gradients or high oscillations, the utilization of exponential or trigonometric polynomial bases confers a superior degree of efficiency than algebraic polynomial bases.…”