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In this work, new finite difference schemes are presented for dealing with the upper-convected time derivative in the context of the generalized Lie derivative. The upper-convected time derivative, which is usually encountered in the constitutive equation of the popular viscoelastic models, is reformulated in order to obtain approximations of second-order in time for solving a simplified constitutive equation in one and two dimensions. The theoretical analysis of the truncation errors of the methods takes into account the linear and quadratic interpolation operators based on a Lagrangian framework. Numerical experiments illustrating the theoretical results for the model equation defined in one and two dimensions are included. Finally, the finite difference approximations of second-order in time are also applied for solving a two-dimensional Oldroyd-B constitutive equation subjected to a prescribed velocity field at different Weissenberg numbers.
We consider finite difference approximations of the second derivative, exemplified in Poissons equation, the heat equation and the wave equation. The finite difference operators satisfy a summation-by-parts property, which mimics the integration-by-p
In the current work we build a difference analog of the Caputo fractional derivative with generalized memory kernel ($_lambda$L2-1$_sigma$ formula). The fundamental features of this difference operator are studied and on its ground some difference sc
We develop an energy-based finite difference method for the wave equation in second order form. The spatial discretization satisfies a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by the
We propose a novel method to compute a finite difference stencil for Riesz derivative for artibitrary speed of convergence. This method is based on applying a pre-filter to the Grunwald-Letnikov type central difference stencil. The filter is obtained
In this article, a numerical scheme is introduced for solving the fractional partial differential equation (FPDE) arising from electromagnetic waves in dielectric media (EMWDM) by using an efficient class of finite difference methods. The numerical s