3. Numerical Methods |

3.1 Numerical Procedure of the Contour Dynamics Model

3.2 Other Details

3.3 Integral Calculation

## 3.2 Other Details

The following practical procedures have the basis on Dritschel(1989).

< Node redistribution >The nodes on the contour are distributed with a uniform interval initially, but the distribution becomes non-uniform after some time, which gives rise to large error in the calculation of the velocity field, and which requires more calculation time due to unnecessarily crowded nodes.

Therefore, the nodes should be redistributed adequately with a finite time interval. First, the node density

r_{i}is defined using the curvaturek_{i}at the nodex_{i}, which is the curvature of the circle including three pointsx_{i-1},x_{i},x_{i+1}:

(3.4)

The nodex_{i}is fixed as a corner if the angle ofx_{i-1},x_{i},x_{i+1}is under 90 degrees. ( The first node is fixed anyway ). And the nodes are redistributed on the contour between two corners, called a contour segment. The number of the nodes n' in the segment is determined withr_{i};

(3.5)

where [ ] is Gauss' notation.Next, i and p , which satisfy the following equation, are sought.

(3.6)

The position of the new nodex_{j}isx(p) , which is located between old nodesx_{i}andx_{i+1}. Note that the form ofx(p) is determined by the interpolation using a cubic spline.

< Interpolation >The contour between adjacent nodes

x_{i}andx_{i+1}is interpolated using a cubic spline.

(3.7)

where,

(3.8)

Here, the interpolation coefficientsa_{i},b_{i}, andg_{i}are given by the curvaturek_{i}as follows;

(3.9)

wherek_{i}and e_{i}are as follows:

(3.10)

Note that the continuity ofxandk_{i}is assumed at the pointsx_{i}andx_{i+1}.

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