Abstract
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.
Highlights
In this paper, we consider the general two-dimensional (2D) linear partial differential equation in the form a(x, y)uxx + b(x, y)uyy + c(x, y)uxy + p(x, y)ux + q(x, y)uy + r(x, y)u = f (x, y). (1)Here the unknown function u, the variable coefficient functions a, b, c, p, q, r, and the forcing function f are assumed to be continuously differentiable and have the required partial derivatives on Ω
We can clearly see that the blended compact difference (BCD) scheme obtains sixth-order accuracy and gets a more accurate solution than the CCD2 scheme
It is seen that the BCD and CCD2 schemes are almost not influenced by the increase of Reynolds number, while the FOC scheme gradually loses its fourth-order accuracy
Summary
For the general 2D linear second-order equations with variable coefficients and the mixed derivative term like (1), it is impossible to get an explicit fourthorder compact difference scheme with 9 grid points except under certain conditions mentioned above. Equation (25) is the new BCD scheme we developed It involves the values of the unknown function on 9 grid points (like explicit compact difference scheme) and their first- and second-order derivatives on 3 grid points on each coordinate direction (like implicit compact difference scheme). With more than 4 grid points along one direction, the above boundary difference equations are not strictly compact in the traditional sense, but they are necessary to retain the high-order accuracy of the BCD scheme.
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