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<p>In this paper, we study a posteriori error estimates of the implicit Euler method for nonlinear neutral delay differential equations (NDDEs). In view of the weak discontinuous property of the solutions to NDDEs, a posteriori error estimates are of utmost importance in numerically solving this class of equations. The error upper bound depending only on the discretization parameters and the data of the problems is first derived for nonlinear NDDEs and applied to the linear case. Based on these a posteriori error estimates, we further develop a time adaptive algorithm. The numerical implementations are performed with both nonuniform partitions and adaptivity in time. The adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution.</p>

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Residual-Based a Posteriori error Analysis and Adaptive Computation for Implicit Euler Method for Nonlinear Neutral Delay Differential Equations

  • 期刊论文

  • 发表日期：

    2026年1月20日

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  • Qiong Chen,
  • Wansheng Wang,
  • Mengli Mao,
  • Chengyu Jin

摘要

In this paper, we study a posteriori error estimates of the implicit Euler method for nonlinear neutral delay differential equations (NDDEs). In view of the weak discontinuous property of the solutions to NDDEs, a posteriori error estimates are of utmost importance in numerically solving this class of equations. The error upper bound depending only on the discretization parameters and the data of the problems is first derived for nonlinear NDDEs and applied to the linear case. Based on these a posteriori error estimates, we further develop a time adaptive algorithm. The numerical implementations are performed with both nonuniform partitions and adaptivity in time. The adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution.