Error Estimates of Scalar Auxiliary Variable Schemes with Variable Time Steps for the Magneto-Hydrodynamic Equations
-
Yuanyuan Mu
,
Hongen Jia
Abstract
In this paper, we employ an adaptive time-stepping strategy to conduct a comprehensive theoretical analysis and numerical simulations of the magnetohydrodynamic (MHD) system. Our objective is to formulate a numerical scheme based on the variable time-steps second-order backward difference formula (VBDF2), which exhibits linear, decoupled, and semi-discrete properties. Firstly, considering the complexity of handling the nonlinear terms in the MHD equations, we will introduce a method that combines the exponential scalar auxiliary variable with the zero-energy contribution. This allows us to construct a new auxiliary variable, which is of crucial importance in dealing with the nonlinear terms. Subsequently, by reformulating the MHD system using this new auxiliary variable, we have successfully derived an equivalent system that is unconditionally stable in terms of energy. Secondly, based on the adaptive time-stepping strategy, we establish a semi-discrete MHD system to guarantee the stability of the system. It is worth noting that, under the favorable condition of being unrestricted by the time-step size, we present a rigorous error analysis. Finally, we carry out several numerical simulations to verify the theoretical results by using an effective adaptive time-step strategy.
Funding source: Shanxi Scholarship Council of China
Award Identifier / Grant number: 2021-029
Funding source: Natural Science Foundation of Shanxi Province
Award Identifier / Grant number: 202203021212249
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12301556
Award Identifier / Grant number: 12426105
Funding statement: This work was supported by the Shanxi Scholarship Council of China (2021-029), Shanxi Province Natural Science Research (202203021212249), National Natural Science Foundation of China (12301556), NSFC Tianyuan Fund-Northwest Center Booster Program for Mathematics (12426105), International Cooperation Base and Platform Project of Shanxi Province (202104041101019) and Basic Research Plan of Shanxi Province (202203021211129).
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