Accuracy Enhancing Interface Treatment Algorithm: The Back and Forth Error Compensation and Correction Method

Abstract

The accuracy of information transmission while solving domain decomposed problems is crucial to the smooth transition of a solution around the interface/overlapping region. This paper describes a systematical study on an accuracy-enhancing interface treatment algorithm based on the back and forth error compensation and correction method (BFECC). By repetitively employing a low order interpolation technique (usually local 2nd order) 3 times, this algorithm achieves local 3rd order accuracy. Analytical derivation for any dimensions is made, and the “superconvergence” phenomenon (4th order accuracy) is found for specific positioning of the source and target grids. Its limitation has also been studied. With rotated grids or grids of different sizes, this method can still reduce the error but fails to improve the order of the underlying interpolation technique. A series of numerical experiments on meshes with various translations, rotations, spacing ratios, and perturbations have been tested. Different interface treatments applied to cavity flow are compared. The 3D flow example shows that the BFECC method positively impacts the convergence of the domain decomposed problem.

Appendices

Appendices

1.1 Comparison of Coefficients Between BFECC and 3rd Order Polynomial Interpolation in 1D

The third-degree Lagrange interpolation polynomial is

$$\begin{aligned} l(x) =&\frac{(x-x_i)(x-x_{i+1})(x-x_{i+2})}{(-\Delta x)(-2\Delta x)(-3\Delta x)}f_{i-1} + \frac{(x-x_{i-1})(x-x_{i+1})(x-x_{i+2})}{\Delta x(-2\Delta x)(-3\Delta x)}f_i \\&+ \frac{(x-x_{i-1})(x-x_i)(x-x_{i+2})}{2\Delta x\cdot \Delta x\cdot (-3\Delta x)}f_{i+1} + \frac{(x-x_{i-1})(x-x_i)(x-x_{i+1})}{3\Delta x\cdot 2\Delta x\cdot \Delta x}f_{i+2}~, \end{aligned}$$

which gives

$$\begin{aligned} l(x_i+\alpha \Delta x)= & {} -\left( \frac{\alpha }{3}-\frac{\alpha ^2}{2}+\frac{\alpha ^3}{6}\right) f_{i-1} + \left( 1-\frac{\alpha }{2}-\alpha ^2+\frac{\alpha ^3}{2}\right) f_i \nonumber \\&+ \left( \alpha +\frac{\alpha ^2}{2}-\frac{\alpha ^3}{2}\right) f_{i+1} - \left( \frac{\alpha }{6}-\frac{\alpha ^3}{6}\right) f_{i+2}~, \end{aligned}$$

(33)

for $0<\alpha <1$.

BFECC interpolation of f at $x_i+\alpha \Delta x$:

(1) Forward:

$$\begin{aligned} f^*_{j-1}&= \alpha f_i + (1-\alpha )f_{i-1}~, \\ f^*_{j}&= \alpha f_{i+1} + (1-\alpha )f_i~, \\ f^*_{j+1}&= \alpha f_{i+2} + (1-\alpha )f_{i+1}~. \end{aligned}$$

(2) Backward:

$$\begin{aligned} {\widetilde{f}}_i&= \alpha f^*_{j-1} + (1-\alpha )f^*_j = \alpha ^2 f_i + \alpha (1-\alpha )f_{i-1} + \alpha (1-\alpha )f_{i+1} + (1-\alpha )^2f_i~, \\ {\widetilde{f}}_{i+1}&= \alpha f^*_j + (1-\alpha )f^*_{j+1} = \alpha ^2 f_{i+1} + \alpha (1-\alpha )f_i + \alpha (1-\alpha )f_{i+2} + (1-\alpha )^2f_{i+1}~. \end{aligned}$$

(3) Compensation:

$$\begin{aligned} {\hat{f}}_i&= f_i + \frac{1}{2}(f_i - {\widetilde{f}}_i) = (1+\alpha -\alpha ^2)f_i - \frac{\alpha }{2}(1-\alpha )f_{i-1} - \frac{\alpha }{2}(1-\alpha )f_{i+1}~, \\ {\hat{f}}_{i+1}&= f_{i+1} + \frac{1}{2}(f_{i+1} - {\widetilde{f}}_{i+1} ) = (1+\alpha -\alpha ^2)f_{i+1} - \frac{\alpha }{2}(1-\alpha )f_i - \frac{\alpha }{2}(1-\alpha )f_{i+2}~. \end{aligned}$$

(4) Forward:

$$\begin{aligned} f^{new}&= \alpha {\hat{f}}_{i+1} + (1-\alpha ) {\hat{f}}_i \\&= -\left( \frac{\alpha }{2}-\alpha ^2+\frac{\alpha ^3}{2}\right) f_{i-1} + \left( 1-\frac{5}{2}\alpha ^2+\frac{3}{2}\alpha ^3\right) f_i \\&\quad + \left( \frac{\alpha }{2}+2\alpha ^2-\frac{3}{2}\alpha ^3\right) f_{i+1} - \left( \frac{\alpha ^2}{2}-\frac{\alpha ^3}{2}\right) f_{i+2}~. \end{aligned}$$

This looks different from (33) (note that they also have different orders of approximation in general.) If we let $\alpha =0.5$, both formulas become

$$\begin{aligned} -\frac{1}{16}f_{i-1} + \frac{9}{16}f_i + \frac{9}{16}f_{i+1} -\frac{1}{16}f_{i+2}~, \end{aligned}$$

which is the ’superconvergence’ case of BFECC interpolation.

1.2 Comparison of BFECC Interpolation at Cell Center with Cubic Least Squares and Bicubic Interpolations in 2D

Consider the cubic least squares interpolation using the values of a sufficiently smooth function at the following 16 nodes, see Fig. 11.

$$\begin{aligned}&g1(0,0), g2(1,0), g3(2,0), g4(3,0) \\&g5(0,1), g6(1,1), g7(2,1), g8(3,1) \\&g9(0,2), g10(1,2), g11(2,2), g12(3,2) \\&g13(0,3),g14(1,3), g15(2,3), g16(3,3) \end{aligned}$$

Fig. 11

Interpolation coefficients and node index

Let a third-degree polynomial in 2D be written as

$$\begin{aligned} P(x,y) = c_1+c_2x+c_3y+c_4x^2+c_5xy+c_6y^2+c_7x^3+c_8x^2y+c_9xy^2+c_{10}y^3 \end{aligned}$$

which is determined by the 16 values of the function by least squares fitting. The coefficients can be obtained in Matlab with symbolic computation. We present the formula at the center as follows.

$$\begin{aligned} \begin{aligned} P(1.5, 1.5) =&-\frac{3}{32}(g_1+g_4+g_{13}+g_{16}) \\&+\frac{1}{16}(g_2+g_3+g_5+g_8+g_9+g_{12}+g_{14}+g_{15}) \\&+\frac{7}{32}(g_6+g_7+g_{10}+g_{11})~. \end{aligned} \end{aligned}$$

(34)

The 2D BFECC interpolation at the center point can be described as follows. For simplicity, we use labels of nodes to represent interpolation values of the function at corresponding nodes at different stages of the method.

(1) Forward:

$$\begin{aligned} f_1^*&= \frac{1}{4}(g_1+g_2+g_5+g_6), \\ f_2^*&= \frac{1}{4}(g_2+g_3+g_6+g_7) \\ f_3^*&= \frac{1}{4}(g_3+g_4+g_7+g_8) \\ f_4^*&= \frac{1}{4}(g_5+g_6+g_9+g_{10}) \\ f_5^*&= \frac{1}{4}(g_6+g_7+g_{10}+g_{11}) \\ f_6^*&= \frac{1}{4}(g_7+g_8+g_{11}+g_{12}) \\ f_7^*&= \frac{1}{4}(g_9+g_{10}+g_{13}+g_{14}) \\ f_8^*&= \frac{1}{4}(g_{10}+g_{11}+g_{14}+g_{15}) \\ f_9^*&= \frac{1}{4}(g_{11}+g_{12}+g_{15}+g_{16}) \\ \end{aligned}$$

(2) Backward:

$$\begin{aligned} {\widetilde{g}}_6&= \frac{1}{4}(f_1^*+f_2^*+f_4^*+f_5^*) \\ {\widetilde{g}}_7&= \frac{1}{4}(f_2^*+f_3^*+f_5^*+f_6^*) \\ {\widetilde{g}}_{10}&= \frac{1}{4}(f_4^*+f_5^*+f_7^*+f_8^*) \\ {\widetilde{g}}_{11}&= \frac{1}{4}(f_5^*+f_6^*+f_8^*+f_9^*) \\ \end{aligned}$$

(3) Compensation:

$$\begin{aligned} {\hat{g}}_6&= g_6 + \frac{1}{2}(g_6 - {\widetilde{g}}_6) \\ {\hat{g}}_7&= g_7 + \frac{1}{2}(g_7 - {\widetilde{g}}_7) \\ {\hat{g}}_{10}&= g_{10} + \frac{1}{2}(g_{10} - {\widetilde{g}}_{10}) \\ {\hat{g}}_{11}&= g_{11} + \frac{1}{2}(g_{11} - {\widetilde{g}}_{11}) \end{aligned}$$

(4) Forward:

$$\begin{aligned} {f}_5^{new} = \frac{1}{4}({\hat{g}}_6+{\hat{g}}_7+{\hat{g}}_{10}+{\hat{g}}_{11}) \end{aligned}$$

Through a direct calculation, we have

$$\begin{aligned} \begin{aligned} {f}_5^{new} =&- \frac{1}{128}(g_1+g_4+g_{13}+g_{16}) \\&-\frac{3}{128}(g_2+g_3+g_5+g_8+g_9+g_{12}+g_{14}+g_{15}) \\&+ \frac{39}{128}(g_6+g_7+g_{10}+g_{11})~. \end{aligned} \end{aligned}$$

(35)

Formula (35) differs from the cubic least squares formula (34), although both formulas have fourth order of accuracy and use the same surrounding nodes in interpolation.

Finally, the bicubic interpolation at the center (1.5, 1.5) can be written as

$$\begin{aligned} \begin{aligned} P_{bic}(1.5,1.5) =&+\frac{1}{256}(g_1+g_4+g_{13}+g_{16}) \\&-\frac{9}{256}(g_2+g_3+g_5+g_8+g_9+g_{12}+g_{14}+g_{15}) \\&+ \frac{81}{256}(g_6+g_7+g_{10}+g_{11})~, \end{aligned} \end{aligned}$$

(36)

which is also different from formula (35).

1.3 Comparison of BFECC Interpolation at Cell Center and Cubic Least Squares Interpolation in 3D

For the least squares method using 64 nodes, we label the nodes similarly as in the 2D case for $z=0$:

$$\begin{aligned}&g1(0,0,0), g2(1,0,0), g3(2,0,0), g4(3,0,0) \\&g5(0,1,0), g6(1,1,0), g7(2,1,0), g8(3,1,0) \\&g9(0,2,0), g10(1,2,0), g11(2,2,0), g12(3,2,0) \\&g13(0,3,0),g14(1,3,0), g15(2,3,0), g16(3,3,0) \end{aligned}$$

For $z=1$, just add 16 to the number label of each node, similarly for $z=2$ and 3. The node indices in the planes of $z=1,2,3$ are varied from $17-32$, $33-48$, $49-64$ respectively.

Let a third-degree polynomial in 3D be written as

$$\begin{aligned} P(x,y,z) =&c_1+c_2x+c_3 y+c_4z \\&+c_5x^2+c_6y^2+c_7z^2+c_8xy+c_9yz+c_{10}zx \\&+c_{11}x^3+c_{12}y^3+c_{13}z^3+c_{14}x^2y+c_{15}x^2z \\&+c_{16}y^2x+c_{17}y^2z+c_{18}z^2x+c_{19}z^2y+c_{20}xyz~. \end{aligned}$$

The coefficients $c1,c2\ldots ,c20$ are calculated by least squares fitting using 64 nodal values. Then we can find its expression at the center point as follows.

$$\begin{aligned} \begin{aligned} P(1.5, 1.5, 1.5) =&-\frac{11}{256}(g_1+g_4+g_{13}+g_{16}+g_{49}+g_{52}+g_{61}+g_{64}) \\&-\frac{1}{256} (g_2+g_3+g_5+g_8+g_9+g_{12}+g_{14}+g_{15}+g_{17}+g_{20} \\&+g_{29}+g_{32}+g_{33}+g_{36}+g_{45}+g_{48}+g_{50}+g_{51}+g_{53}+g_{56}+g_{57}\\&+g_{60}+g_{62}+g_{63})+\frac{9}{256} (g_6+g_7+g_{10}+g_{11}+g_{18}\\&+g_{19}+g_{21}+g_{24}+g_{25}+g_{28}+g_{30}+g_{31}+g_{34}+g_{35}\\&+g_{37}+g_{40}+g_{41}+g_{44}+g_{46}+g_{47}+g_{54}+g_{55}+g_{58}+g_{59})\\&+\frac{19}{256}(g_{22}+g_{23}+g_{26}+g_{27}+g_{38}+g_{39}+g_{42}+g_{43}). \end{aligned} \end{aligned}$$

(37)

For 3D BFECC interpolation at the center point, we can get the expression as follows (the process is very similar to the 2D case in Appendix B and skipped here.)

$$\begin{aligned} \begin{aligned} f_{14}^{new} =&-\frac{1}{1024}(g_1+g_4+g_{13}+g_{16}+g_{49}+g_{52}+g_{61}+g_{64}) \\&-\frac{3}{1024} (g_2+g_3+g_5+g_8+g_9+g_{12}+g_{14}+g_{15}+g_{17}+g_{20}+g_{29}+g_{32} \\&+g_{33}+g_{36}+g_{45}+g_{48}+g_{50}+g_{51}+g_{53}+g_{56}+g_{57}+g_{60}+g_{62}+g_{63})\\&-\frac{9}{1024} (g_6+g_7+g_{10}+g_{11}+g_{18}+g_{19}+g_{21}+g_{24}+g_{25}+g_{28}+g_{30}+g_{31}\\&+g_{34}+g_{35}+g_{37}+g_{40}+g_{41}+g_{44}+g_{46}+g_{47}+g_{54}+g_{55}+g_{58}+g_{59})\\&+\frac{165}{1024}(g_{22}+g_{23}+g_{26}+g_{27}+g_{38}+g_{39}+g_{42}+g_{43})~. \end{aligned} \end{aligned}$$

(38)

Formula (38) is different from formula (37), even though both formulas have fourth order of accuracy and use the same surrounding nodes in interpolation.