Fourth‐order compact scheme based on quasi‐variable mesh for three‐dimensional mildly nonlinear stationary convection–diffusion equations

Abstract

A new family of compact schemes of increased accuracy using quasi-variable mesh is presented for determining approximate solutions to the three-space dimensions mildly nonlinear convection dominated diffusion equations. The main thought behind the proposed scheme is to get uniformly distributed local truncation error, which otherwise not possible in case of finite-difference discretization using constant step-sizes mesh points. According to the zero or nonzero values of mesh stretching quantities, the increased accuracy fourth-order method refers to uniform meshes or quasi-variable meshes schemes. It is easy to tune the mesh points depending upon the location of subdomains having comparatively more fluctuating solution behavior. The block-tridiagonal construction of the Jacobian (iteration matrix) acquired with the new difference scheme makes it easier to implement with a reasonable computing time. We will describe the matrix and graph theoretic approach to analyze the convergence property and error estimate of the generalized scheme. A measure of accuracy, such as maximum errors, root-mean-squared errors, and numerical convergence rate, is examined by solving various forms of convection-dominated diffusion equations.

Fuzzy transform algorithm based on high-resolution compact discretization for three-dimensional nonlinear elliptic PDEs and convection-diffusion equations

“…Fig.2presents the surface plot of the solution 𝑢(𝑥, 𝑦, 𝑧) at 𝑥 = -1 and Fig.3illustrate the sliced contour plot of solution for four fixed values of 𝑥 = -1, -0.8, -0.6, -0.4 by taking 𝐾 = 20. The four quadrants of contour indicate the higher magnitude of depression and elevation for which classical solution techniques remain unstable.Example 7.3Jha and Singh (2020) Consider the linear Helmholtz equation 𝜋 2 (𝑎 2 + 𝑏 2 + 𝑐 2 )𝑢 = 𝜓(𝑥, 𝑦, 𝑧).…”