VỀ TÍNH ỔN ĐỊNH CỦA PHƯƠNG PHÁP DỰ BÁO - HIỆU CHỈNH HỌ ADAMS VÀ HỌ SAI PHÂN LÙI

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Tác giả: Dinh Van Tiep, Thi Thu Hang Pham

Ngôn ngữ: vie

Ký hiệu phân loại:

Thông tin xuất bản: Tạp chí Khoa học và Công nghệ - Đại học Thái Nguyên, 2020

Mô tả vật lý: tr.24-30

Bộ sưu tập: Metadata

ID: 313465

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Phương pháp dự báo - hiệu chỉnh có ưu điểm trong việc giảm đáng kể số lượng tính toán giá trị hàm số và đạo hàm so sánh với phương pháp đơn bước kiểu Runge-Kutta truyền thống cũng như so với nhiều phương pháp đa bước khác. Tính ổn định là một vấn đề truyền thống đối với các phương pháp khi xét ở bậc cao. Bài báo này đề cập đến tính ổn định, và so sánh chúng, của phương pháp với k-bước dự báo kiểu Adams-Brashfort và (k+1)-bước hiệu chỉnh kiểu Adams-Moulton hay kiểu sai phân lùi (BDF) với Lý do đề cập đến kiểu hiệu chỉnh BDF ở đây có nguồn gốc từ thực tế rằng một hiệu chỉnh BDF tạo ra một miền ổn định tuyệt đối lớn, với so với hiệu chỉnh kiểu Adams-Moulton. Một số nhược điểm của các phương pháp dự báo - hiệu chỉnh này khi áp dụng cho các bài toán stiff cũng được đề cập trong bài báo cùng với một thuật toán cải tiến cho các phương pháp này được phát triển để khắc phục phần nào nhược điểm đó. Đóng góp lớn nhất của bài báo là phương pháp xây dựng đa thức ổn định dựa vào phương trình sai phân mô tả phương pháp dự đoán và phương pháp hiệu chỉnh. Dựa vào đa thức này, phương pháp tập hợp đường bao được áp dụng để mô tả trực quan miền ổn định của phương pháp.The predictor-corrector methods take the upper hand in decreasing the number of function evaluations and of derivative evaluations as well comparing to the Runger-Kutta methods and various linear multistep methods. The stability however is a traditional disease of a high order method. This problem is also the case to the predictor-corrector method. The paper discusses on the matter of stability of the k-step Adams predictor-correct method and that of the predictor-corrector methods constructed on the basis of k-step Adams-Brashfort for the predictor and the k-step or (k+1)-step backward difference formula (BDF) for the corrector with low k, say . The reason to consider the BDF corrector is from the fact of having a large portion of the absolute stability region for those methods (with ) competing to other Adams-Moulton correctors. Some awkward performances of the predictor-corrector to the stiffness are also discussed and a modified algorithm is also developed to treat the poor performance of the abovementioned methods. The main contribution of the paper is the strategy of depicting the absolute stability region of a predictor-corrector method by constructing its stability polynomial on the basis of the recurrence equation obtain form the pair of difference equations describing the predictor and corrector. On that construction, we can be able to sketch the region by the boundary locus method.

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