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.

A higher-order blended compact difference (BCD) scheme is proposed to solve the general two-dimensional (2D) linear second-order partial differential equation. The distinguishing feature of the present method is that methodologies of explicit compact difference and implicit compact difference are blended together. Sixth-order accuracy approximations for the first-and second-order derivatives are employed, and the original equation is also discretized based on a 9-point stencil, which is different from the work of Lee et al. (J. Comput. Appl. Math. 264:23-37, 2014). A truncation error analysis is performed to show that the scheme is of sixth-order accuracy for the interior grid points. Simultaneously, sixth-order accuracy schemes are proposed to compute the grid points on the boundaries for the first-and second-order derivatives. Numerical experiments are conducted to demonstrate the accuracy and efficiency of the present method.

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In this study, we present a fourth-order and a sixth-order blended compact difference (BCD) schemes for approximating the three-dimensional (3D) convection–diffusion equation with variable convective coefficients. The proposed schemes, where transport variable, its first and second derivatives are carried as the unknowns, combine virtues of compact discretization, fourth-order Padé scheme and sixth-order combined compact difference (CCD) scheme for spatial derivatives and can efficiently capture numerical solutions of linear and nonlinear convection–diffusion equations with Dirichlet boundary conditions. The fourth-order scheme requires only 7 grid points and the sixth-order scheme requires 19 grid points. The distinguishing feature of the present method is that methodologies of explicit compact difference and implicit compact difference are blended together. The truncation errors of the two difference schemes are analyzed for the interior grid points, respectively. Simultaneously, a sixth-order accuracy scheme is proposed to compute the first and second derivatives of the grid points on boundaries. Finally, the presented methods are applied to several test problems from the literature including linear and nonlinear problems. It is found that the presented schemes exhibit good performance.

This paper proposes a higher-order blended compact difference (BCD) scheme on nonuniform grids for solving the three-dimensional (3D) convection–diffusion equation with variable coefficients. The BCD scheme has fifth- to sixth-order accuracy and considers the first and second derivatives of the unknown function as unknowns as well. Unlike other schemes that require grid transformation, the BCD scheme does not require any grid transformation and is simple and flexible in grid subdivisions. Concurrently, the corresponding high-order boundary schemes of the first and second derivatives have also been constructed. We tested the BCD scheme on three problems that involve convection-dominated and boundary-layer features. The numerical results show that the BCD scheme has good adaptability and high resolution on nonuniform grids. It outperforms the BCD scheme on uniform grids and the high-order compact scheme on nonuniform grids in the literature in terms of accuracy and resolution.