非线性抛物型积分-微分方程的高阶标量辅助变量方法

doi:10.13371/j.cnki.chin.q.j.m.2025.03.003

数学季刊 ›› 2025, Vol. 40 ›› Issue (3): 262-270.doi: 10.13371/j.cnki.chin.q.j.m.2025.03.003

非线性抛物型积分-微分方程的高阶标量辅助变量方法

1. School of Mathematics and Statistics, Henan University, Kaifeng 475004, China;
2. School of Mathematics and Statistics, Yunnan University, Kunming 650504, China;
3. Center for Applied Mathematics of Henan Province, Henan University, Zhengzhou 450046, China

收稿日期:2025-05-09 出版日期:2025-09-30 发布日期:2025-09-30
作者简介:YAN Li-na (2003-), female, native of Xinxiang, Henan, graduate student of Henan University; ZHANG Gen-gen (1986-), male, native of Jiujiang, Jiangxi, associate professor of Yunnan University, engages in numerical methods of PDEs; HUANG Qiong-ao (1990-), male, native of Zhoukou, Henan, associate professor of Henan University, engages in numerical methods of PDEs.
基金资助:
Supported by the National Natural Science Foundation of China (Grant Nos. 12001210 and 12261103), the Natural Science Foundation of Henan (Grant No. 252300420308) and the Yunnan Fundamental Research Projects (Grant No. 202301AT070117).

A High-Order Scalar Auxiliary Variable Approach for Nonlinear Parabolic Integro-Differential Equations

1. School of Mathematics and Statistics, Henan University, Kaifeng 475004, China;

2. School of Mathematics and Statistics, Yunnan University, Kunming 650504, China;

3. Center for Applied Mathematics of Henan Province, Henan University, Zhengzhou 450046, China

Received:2025-05-09 Online:2025-09-30 Published:2025-09-30
About author:YAN Li-na (2003-), female, native of Xinxiang, Henan, graduate student of Henan University; ZHANG Gen-gen (1986-), male, native of Jiujiang, Jiangxi, associate professor of Yunnan University, engages in numerical methods of PDEs; HUANG Qiong-ao (1990-), male, native of Zhoukou, Henan, associate professor of Henan University, engages in numerical methods of PDEs.
Supported by:
Supported by the National Natural Science Foundation of China (Grant Nos. 12001210 and 12261103), the Natural Science Foundation of Henan (Grant No. 252300420308) and the Yunnan Fundamental Research Projects (Grant No. 202301AT070117).

摘要/Abstract

摘要： An efficient and accurate scalar auxiliary variable (SAV) scheme for numerically solving nonlinear parabolic integro-differential equation (PIDE) is developed in this paper. The original equation is first transformed into an equivalent system, and the k-order backward differentiation formula (BDFk) and central difference formula are used to discretize the temporal and spatial derivatives, respectively. Different from the traditional discrete method that adopts full implicit or full explicit for the nonlinear integral terms, the proposed scheme is based on the SAV idea and can be treated semi-implicitly, taking into account both accuracy and effectiveness. Numerical results are presented to
demonstrate the high-order convergence (up to fourth-order) of the developed schemes and it is computationally efficient in long-time computations.

关键词: Parabolic integro-differential equation, Scalar auxiliary variable, Fredholm equation, High-order BDF scheme

Abstract: An efficient and accurate scalar auxiliary variable (SAV) scheme for numerically solving nonlinear parabolic integro-differential equation (PIDE) is developed in this paper. The original equation is first transformed into an equivalent system, and the k-order backward differentiation formula (BDFk) and central difference formula are used to discretize the temporal and spatial derivatives, respectively. Different from the traditional discrete method that adopts full implicit or full explicit for the nonlinear integral terms, the proposed scheme is based on the SAV idea and can be treated semi-implicitly, taking into account both accuracy and effectiveness. Numerical results are presented to
demonstrate the high-order convergence (up to fourth-order) of the developed schemes and it is computationally efficient in long-time computations.

Key words: Parabolic integro-differential equation, Scalar auxiliary variable, Fredholm equation, High-order BDF scheme

中图分类号:

O241.8

闫丽娜, 张根根, 黄琼敖. 非线性抛物型积分-微分方程的高阶标量辅助变量方法[J]. 数学季刊, 2025, 40(3): 262-270.

YAN Li-na, ZHANG Gen-gen, HUANG Qiong-ao. A High-Order Scalar Auxiliary Variable Approach for Nonlinear Parabolic Integro-Differential Equations[J]. Chinese Quarterly Journal of Mathematics, 2025, 40(3): 262-270.

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非线性抛物型积分-微分方程的高阶标量辅助变量方法

A High-Order Scalar Auxiliary Variable Approach for Nonlinear Parabolic Integro-Differential Equations

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