TAILIEUCHUNG - CHL - A Finite Element Scheme for Shock Capturing_4

Tham khảo tài liệu 'chl - a finite element scheme for shock capturing_4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Simpo PDF Merge and Split Unregistered Version - http one moves over time the center-line profile shock moves upstream. It is apparent that as the spatial and temporal resolution improve the shock becomes steeper. The shock is fairly consistently spread over three or four elements and so as the element size is reduced the resulting shock is steeper. The x-t slope of the shock indicates the shock speed. Any bending would indicate that the speed changed over time which should not be the case. The upper elevation is precisely m which is correct. There is no overshoot of the jump though there is some undershoot when Cs is less than 1. Cs is the product of the analytic shock speed and the ratio of time-step length to element length. A Cs value of 1 indicates that the shock should move 1 element length in 1 time-step. Figures 12 and 13 show the error in calculated speed and the relative error in calculated speed respectively. These are for AY and m which is reflected in the Grid Resolution Number defined as XX AJi. Here h is the depth and Ah is the analytic depth difference across the shock m. The error was as small as was detectable by the technique for measurement of speed at AX m so there was no need to go to smaller grid spacing. Values of Cs less than 1 appear to lag the analytic shock and Cs greater than 1 leads the analytic shock. With the largest Cs the calculated shock speed is greater than the analytic by at most m sec which is only percent too fast. As resolution is improved the solution appears to converge to the analytic speed. Figures 14-16 and 17-19 are the center-line profile histories for at and for AX and m respectively. It is apparent that the lower dissipation from this second-order scheme allows an oscillation which is most notable upstream of the jump for larger values of Cs. But as Cs decreases there is an undershoot in front of the shock. The slope of the x-t line along the top of the

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