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کاربست چندجملهایهای هرمیت مکعبی در ساخت روش نیمهلاگرانژی یکنوا برای معادله فرارفت | ||
| فیزیک زمین و فضا | ||
| مقاله 8، دوره 45، شماره 1، فروردین 1398، صفحه 107-127 اصل مقاله (900.95 K) | ||
| نوع مقاله: مقاله پژوهشی | ||
| شناسه دیجیتال (DOI): 10.22059/jesphys.2019.253532.1006982 | ||
| نویسندگان | ||
| علی محمدی1؛ علیرضا محب الحجه* 2؛ مجید مزرعه فراهانی3 | ||
| 1دانشجوی دکتری، گروه فیزیک فضا، مؤسسه ژئوفیزیک، دانشگاه تهران، تهران، ایران | ||
| 2استاد، گروه فیزیک فضا، مؤسسه ژئوفیزیک، دانشگاه تهران، تهران، ایران | ||
| 3دانشیار، گروه فیزیک فضا، مؤسسه ژئوفیزیک، دانشگاه تهران، تهران، ایران | ||
| چکیده | ||
| از آنجا که روشهای نیمهلاگرانژی محدودیت پایداری به‌شرط کورانت-فردریکس-لوی (CFL) ندارند، بهصورت وسیعی در مدلهای گردش کلی بهکار برده میشوند. در غالب صورتهای روش نیمهلاگرانژی، برای محاسبه کمیت حلشونده (مانند سرعت باد، رطوبت، دما و جرم) درون‌یابی در فاصله بین نقاط شبکه ای لازم است. از مشکلات مواجه در کاربرد روش نیمهلاگرانژی تولید نوسان اضافی در مناطق با گرادیان شدید است. دو رویکرد برای حذف نوسانهای اضافی در مناطق با گرادیان شدید بهکار گرفته میشود: الف) ترکیب یک روش با مرتبه بالا با روش پادجریانسو، ب) استفاده از روشهای درون‌یابی یکنوا. در این پژوهش از روش درون‌یابی گزینشی با چندجملهایهای هرمیت مکعبی نایکنوا و یکنوا با صافی مشتق برای درون‌یابی مقادیر کمیت حلشونده در فاصله بین نقاط شبکهای استفاده شده است. نتایج نشان میدهد که حل معادله فرارفت با روش مذکور نوسان اضافی در مناطق با گرادیان شدید ایجاد نمیکند و میزان میرایی آن نیز ناچیز است. عدم تولید مقادیر منفی با استفاده از درون‌یاب یکنوای هرمیت در حل میدانهای ذاتاً نامنفی مانند رطوبت و جرم از دیگر نتایج این پژوهش است. | ||
| کلیدواژه‌ها | ||
| فرارفت؛ نیمهلاگرانژی؛ یکنوا؛ چندجمله‌ای هرمیت؛ جرم؛ رطوبت | ||
| عنوان مقاله [English] | ||
| Using cubic Hermite polynomials in constructing monotone semi-Lagrangian methods for advection equation | ||
| نویسندگان [English] | ||
| Ali Mohammadi1؛ Ali Reza Mohebalhojeh2؛ Majid Mazraeh Farahani3 | ||
| 1Ph.D. Student, Department of Space Physics, Institute of Geophysics, University of Tehran, Tehran, Iran | ||
| 2Professor, Department of Space Physics, Institute of Geophysics, University of Tehran, Tehran, Iran | ||
| 3Associate Professor, Department of Space Physics, Institute of Geophysics, University of Tehran, Tehran, Iran | ||
| چکیده [English] | ||
| Semi-Lagrangian methods have been widely applied in general circulation models of the atmosphere as they do not suffer from a Courant–Fredericks–Levy (CFL) constraint for computational stability. Ease of application, high accuracy and speed of execution in general circulation models are other reasons for the popularity of semi-Lagrangian methods. Two fundamenta lissues in semi-Lagrangian methods are related to the trajectory computation and interpolation from the regular grid to departure points. If sufficiently accurate schemes are used to solve for trajectories with interpolations, one can expect good performance from the semi-Lagrangian scheme in solving the equations of motion of the atmosphere. Two general methods of solving the trajectory equation are the forward and backward methods. Most semi-Lagrangian methods use backward-trajectory schemes for estimating positions of the air parcels that arrive at the grid points in the future time step. Solving the trajectory equation is carried out by iteration. In the research reported, two iterations are used for trajectory computation. The fundamental difference between the forward and backward trajectory scheme rests in the calculation of advective quantity at the departure and destination points. While in the backward solution procedure, it is necessary to make interpolation from the regular grid to departure points; in the forward scheme, it is necessary to make interpolation from the irregular grid of destination points to the regular grid. The usually used interpolation methods in the semi-Lagrangian method include piecewise cubic Lagrange and Hermite, cascades, and monotone Hermite. Increasing the degree of polynomial interpolation leads to a higher degree of formal accuracy, but it leads to the generation of unwanted oscillation in regions with severe gradients of the transported quantities. Eliminating the unwanted oscillations is done through a variety of methods which generally increase the computational cost and reduce the accuracy of the scheme. To address the issue, in this research, a new selective monotone semi-Lagrangian method is developed and tested along with two standard methods based on the Lagrange and Hermite interpolations. The Lagrange polynomials have been considered by researchers for the high speed of computation in operational models. The fictitious oscillations produced at the edges of sharp gradients of the advected quantities are the main shortcoming of this method. The fictitious oscillations cannot be eliminated by increasing the degree of interpolation polynomials, which can only lead to a reduction in the wavelength of the oscillations. The results presented on increasing the degree of interpolation polynomials clearly show that the removal of the fictitious oscillations requires the use of monotone polynomials for interpolation. It is important to note that the Hermite interpolation polynomials are not inherently monotone. To make them monotone, one needs to manipulate the derivatives at the grid points appropriately. This process, however, may lead to a substantial deteriration of accuracy. For this reason, in this paper, a selective interpolation method is desined to obtain the best accuracy in solution of the advection equation, while preserving monotnonicity and removing the issue with the fictitious oscillations. In the selective method, first the interpolation is done by the non-monotonic cubic Hermite and then a properly designed slope function is calculated at each grid interval. If the slope function takes values outside the range, it indicates that a fictitious oscillation has occurred in the interpolantion. To remove the oscillation, the non-monotone interpolation is abandoned and the monotone interpolation is performed by limiting the derivative to the monotone region. This technique can minimize the error caused by the changes in the derivatives. Results are shown to demonstrate the working and superiority of the seclective montone scheme. | ||
| کلیدواژه‌ها [English] | ||
| Advection, Semi-Lagrangian, Monotone, Hermite polynomial, Mass, Humidity | ||
| مراجع | ||
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