High-order blended compact difference schemes for the 3D elliptic partial differential equation with mixed derivatives and variable coefficients

Tingfu Ma,Yongbin Ge

doi:10.1186/s13662-020-02949-7

Abstract

Blended compact difference (BCD) schemes with fourth- and sixth-order accuracy are proposed for approximating the three-dimensional (3D) variable coefficients elliptic partial differential equation (PDE) with mixed derivatives. With truncation error analyses, the proposed BCD schemes can reach their theoretical accuracy, respectively, for the interior gird points and require 19 points compact stencil. They fully blend the implicit compact difference (CD) scheme and the explicit CD scheme together to make the derivation method and programming easier. The BCD schemes are also decoupled, which means the unknown function and its derivatives are separately resolved by different finite difference equations. Moreover, the sixth-order schemes are developed to solve the first-order derivatives, the second-order derivatives and the second-order mixed derivatives on boundaries. Several test problems are applied to show that the present BCD schemes are more accurate than those in the literature.

Highlights

In the paper, we study the steady 3D elliptic partial differential equation (PDE) auxx + buyy + cuzz + pux + quy + ruz + d1uxy + d2uyz + d3uzx + su = f (x, y, z), (1)where a, b, c, p, q, r, d1, d2, d3, s and f are sufficiently smooth functions and have the required partial derivatives on
The main aim of the present paper is to extend our research work for 2D elliptic equations [35] to the 3D cases with variable coefficients and mixed derivatives to derive fourth- and sixth-order Blended compact difference (BCD) schemes
The unknown function and its first-order derivatives are regarded as the unknown variables in the calculation

Summary

Introduction

Substituting δx, δy, δz, δx, δy2, δz, δxδy, δyδz, δzδx (see Appendix A) at grid point 0 into (62), and neglecting the truncation error terms, the 19-point sixth-order BCD scheme can be derived as follows: 4A1 3h2x. The mixed derivatives {∂x∂yu, ∂y∂zu, ∂z∂xu} are computed with the nine-point sixth-order schemes in Ref. In order to obtain the sixth-order scheme of the left boundary (i = 0 and i = Nx in x-direction) of ux, we assume that the unknown function and its first-order derivative ux have the following relationship:. All sixth-order boundaries schemes (left, right, down, up, rear and front) for {uxx, uyy, uzz} are given as follows:. We obtain all sixth-order boundaries schemes which require more than 3 grid points They are not compact in the traditional sense, the major compact structure is from the interior difference equations.

Conclusions

Methods