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

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.

Automated summary

The study introduces a new family of compact schemes using quasi-variable mesh to enhance the accuracy of approximate solutions to mildly nonlinear convection dominated diffusion equations. By adjusting mesh points to achieve uniformly distributed local truncation errors, the scheme can be easily tuned based on the fluctuating solution behavior of subdomains. The block-tridiagonal construction of the Jacobian matrix acquired with the new scheme facilitates implementation with reasonable computing time and analysis of error estimates and convergence properties using matrix and graph theoretic approaches.