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Accuracy Analysis of the Immersed Boundary Method Using the Method of Manufactured Solutions

GONG Zhao-xin 1 , LU Chuan-jing 1,2 , HUANG Hua-xiong 3 (1.Department of Mechanics, Shanghai Jiaotong University, Shanghai 200240,P.R.China. 2. State Key Laboratory of Ocean Engineering, Shanghai 200030,P.R.China. 3. Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada)  
The immersed boundary method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics. The principle work was to analyze the accuracy of the immersed boundary method. The whole procedure contained two parts: code verification and accuracy analysis. The code verification part provided the confidence that the code used here was mistake free and the accuracy analysis part gave the order of accuracy of the immersed boundary method. The method of manufactured solutions was taken as the research means for both parts. In the first part, the numerical code employed a second order discretization scheme, i.e., the theoretical order of accuracy was second order. It was matched by the calculated order of accuracy obtained by numerical calculation for all variables. This meant that the code contained no mistake, which was the premise of the following work. The second part introduced a jump in the manufactured solution for pressure and added corresponding singular forcing terms in the momentum equations. By analyzing the discretization errors, the accuracy of the immersed boundary method was proved to be first order, even though the discretization scheme was second order. In the meantime, it was found that the coarser mesh might be not sensitive enough to capture the influence of the immersed boundary and that refinement on the Lagrangian markers barely had any effects on the numerical calculation.
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