This article proposes a noniteration solution based on the Lie-group shooting method (LGSM) to solve the backward-in-time two-dimensional Burgers’ equation with a large Reynolds number. The backward problem is famous for seriously ill-posed cases because the solution is generally unstable and highly dependent on the input data. Small perturbations in the input data, such as random errors inherent to the measurements in the analysis, can cause large oscillations in the solution. To handle a large Reynolds number under long time spans, it is very difficult to integrate towards the time direction. To avoid time integration and numerical iteration, the noniteration vector solution based on a two-point equation of the LGSM, including the initial and final conditions and boundary conditions (BCs) at the initial and terminal times, can be constructed. When the vector solution can be obtained from the ratios of the wave fronts on the BCs at the initial and terminal times, this solver can avoid the numerical iteration and numerical divergence of the conventional LGSM. Two benchmark examples in one and two variables are examined to illustrate the performance of the proposed method. The numerical results of this research are very consistent with the exact solutions when considering disturbances from noisy data. Even when the Reynolds number reaches 10E12, from the noisy final and boundary data, the noniteration solution can efficiently address the nonlinear Burgers’ problem with or without disturbances. This method does not use any transformation techniques, iterative processes, or regularization processes to avoid numerical instability. Hence, a noniterative solution is more stable and accurate for the unsteady nonlinear Burgers’ equation than currently used methods.
Chang, Yen-Shen; Chen, Yung-Wei; Liu, Chein-Shan; and Chang, Jiang-Ren
"A Non-Iteration Solution for Solving the Backward-in-Time Two-Dimensional Burgers’ equation with a Large Reynolds Number,"
Journal of Marine Science and Technology: Vol. 30:
1, Article 7.
Available at: https://jmstt.ntou.edu.tw/journal/vol30/iss1/7