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

YAN Li-na, ZHANG Gen-gen, HUANG Qiong-ao

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

Chinese Quarterly Journal of Mathematics >

2025 , Vol. 40 >Issue 3: 262 - 270

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2025.03.003

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

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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

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.

HUANG Qiong-ao (1990-), male, native of Zhoukou, Henan, associate professor of Henan University, engages in numerical methods of PDEs.

Received date: 2025-05-09

Online published: 2025-09-30

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).

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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

Cite this article

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

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