Functionally-fitted block $ \theta $-methods for ordinary differential equations

Jingwen Wu; Jintao Hu; Hongjiong Tian

doi:10.3934/dcdss.2020164

Article Contents

2020, Volume 13, Issue 9: 2603-2617. Doi: 10.3934/dcdss.2020164

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Functionally-fitted block $ \theta $-methods for ordinary differential equations

1.
College of Information, Mechanical and Electrical Engineering, Shanghai Normal University, 100 Guilin Road, Shanghai 200234, China
2.
Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, 100 Guilin Road, Shanghai 200234, China

^* Corresponding author: Hongjiong Tian
^* Corresponding author: Hongjiong Tian

Dedicated to the memory of Professor Christopher T. H. Baker

Received: December 2018

Revised: May 2019

Early access: December 2019

Published: June 2020

The work of the authors is supported in part by E-Institutes of Shanghai Municipal Education Commission under Grant No. E03004, the National Natural Science Foundation of China under Grant Nos. 11671266 and 11871343, and the Natural Science Foundation of Shanghai under Grant No. 16ZR1424900

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Abstract

We propose a new family of functionally-fitted block $ \theta $-methods for numerically solving ordinary differential equations which integrates a chosen set of linearly independent functions exactly. The advantage of such variable coefficient methods is that the basis functions can be chosen to exploit specific properties of the problem that may be known in advance. The basic theory for the proposed methods is established. First, we derive a sufficient condition to ensure the existence of the functionally-fitted block $ \theta $-methods, and discuss the independence on integration time for a set of separable basis functions. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the order of accuracy of $ r $-dimensional functionally-fitted block $ \theta $-method is at least $ r $ for ordinary differential equations. Numerical experiments are conducted to illustrate the efficiency of the functionally-fitted block $ \theta $-methods.

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Mathematics Subject Classification: Primary: 65L05, 65M10, 65M20; Secondary: 65R20.

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Figure 1. Global errors $ \log_{2}(E) $ of FFBT1, FFBT2, FFBT3 and BT with various $ \varepsilon $

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Figure 2. Global errors $ \log_{10}E $ (left) and Execution times (right) of FFBT and FFB

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Table 1. Maximum absolute errors of FFBT(Ⅰ)-FFBT(Ⅲ), BT(Ⅰ)-BT(Ⅲ)

$ h $	FFBT(Ⅰ)	FFBT(Ⅱ)	FFBT(Ⅲ)	BT(Ⅰ)	BT(Ⅱ)	BT(Ⅲ)
0.4	3.66e-19	4.04e-13	2.07e-01	1.00e-04	2.51e-05	3.16e-01
0.2	6.30e-19	2.87e-18	3.20e-03	2.51e-05	5.01e-06	1.58e-06
0.1	2.74e-18	1.69e-18	2.41e-11	5.01e-06	1.26e-06	3.98e-07
0.05	1.11e-17	9.41e-18	2.91e-18	1.26e-06	3.16e-07	1.00e-07

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Table 2. Maximum absolute errors of FFBT(A)-FFBT(B), BT(A)-BT(B).

$ h $	FFBT(A)	FFBT(B)	BT(A)	BT(B)
0.4	2.93e-04	1.39e-04	5.92e-02	1.41e-04
0.2	1.12e-04	6.58e-05	6.35e-02	6.60e-05
0.1	5.72e-05	4.46e-05	6.57e-02	4.47e-05
0.05	2.86e-05	1.11e-05	6.68e-02	1.11e-05

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