**3.3 Improving the accuracy of circuits using the Richardson extrapolation method**

The Richardson extrapolation method is used to solve grid problems on a sequence of grids. The method consists in carrying out calculations for the same circuit, with different steps. Then we have several grid solutions. On the basis of the grid solutions, some linear combination is compiled. The resulting linear combination has a higher order of accuracy.

Creation of new schemes using Richardson extrapolation based on the schemes given in paragraph 3.1.

The accuracy of scheme (76), with a uniform arrangement of grid nodes, is *O h*ð Þ. Scheme (79) has order *O h*ð Þ *=*2 . For a linear combination

*Q*3 ð Þ¼� *xi* 1 <sup>3</sup> *<sup>U</sup>*<sup>1</sup> ð Þþ *xi* 4 <sup>3</sup> *<sup>U</sup>*<sup>3</sup> ð Þ *xi* , we get an approximation error for a uniform grid *O h*<sup>2</sup> � �. A linear combination of *U*<sup>1</sup> ð Þ *xi* , *<sup>U</sup>*<sup>3</sup> ð Þ *xi* and *<sup>U</sup>*<sup>7</sup> ð Þ *xi* in the form *Q*<sup>7</sup> ð Þ¼ *xi* 1 <sup>45</sup> *<sup>U</sup>*<sup>1</sup> ð Þ� *xi* 4 <sup>9</sup> *<sup>U</sup>*<sup>3</sup> ð Þþ *xi* 64 <sup>45</sup> *<sup>U</sup>*<sup>7</sup> ð Þ *xi* has an approximation order of *O h*<sup>4</sup> � �. Consider *<sup>N</sup>* <sup>¼</sup> 10, *S x*ð Þ¼ *<sup>x</sup>*2, *Pe* <sup>¼</sup> 30. **Table 2** shows the absolute difference between the exact and approximate solutions according to the schemes.

**Table 3** shows the root-mean-square error *σ* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P*<sup>N</sup>* <sup>1</sup> ð Þ *Ф*ð Þ� *xi Ui* 2 *=N* q for the considered schemes. *Ф*ð Þ *xi* the exact solution at the nodal points, *Ui* is the numerical solution obtained by the considered schemes.

**Figures 41** and **42** show numerical solutions for Ф<sup>W</sup> = 0, Ф<sup>E</sup> = 0.

From the graphs in **Figures 41** and **42**, and from **Tables 2** and **3**, it is clear that the Richardson linear combination allows you to get a more improved circuit.


**Table 2.**

*The absolute difference between the exact and approximate solutions.*


#### **Table 3.**

*Comparison by the root-mean-square errors.*

#### **Figure 41.**

*Pe = 100, S = 5cos4πx Solid curve exact solution, circle obtained by scheme U<sup>1</sup> , circle by U3 , solid rectangle by U7 , diamond by Q<sup>3</sup> , star by Q7 .*

#### **Figure 42.**

*Pe = 100, S = exp.(4x). Solid curve exact solution, circle obtained by scheme U1 , circle by U<sup>3</sup> , solid rectangle by U7 , diamond by Q<sup>3</sup> , star by Q<sup>7</sup> .*
