A Meshfree Method for Wave Propagation using the Multi-variable Taylor Expansion

Abstract:

We applied a novel mesh-free method to solve wave equations. Although the conventional finite difference methods determine the coefficients of its operator based on the regular grid alignment, the mesh-free method is not restricted to regular arrangements of nodes. We derive the mesh-free approach using the multi-variable Taylor expansion. The methodology can provide arbitrary-order accuracy in space by expanding the influence domain which controls the number of neighboring nodes. First, the dispersion property of the method is investigated using a plane wave analysis. Although the scheme for the irregular distribution of the calculation points is more dispersive than that of the regular alignment, the choice of the higher-order scheme can improve the dispersion property of the method. Next, we demonstrate the effectiveness of the method using homogeneous and heterogeneous media. In the homogeneous medium case, we calculate acoustic wave propagation with regular and irregular distributions of the calculation points. The result from the irregular arrangement has good agreement with that of the regular one. In the heterogeneous medium case, we set a low velocity anomaly in the model, and calculate acoustic wave propagation with fine, coarse and partially fine arrangements of the calculation points. The coarse arrangement induces severe numerical dispersion inside the low velocity anomaly. On the other hand, partially fine arrangement can suppress the numerical dispersion, and can reproduce accurate wave field as well as the fine arrangement. Since the partially fine arrangement use fine alignment only around the anomaly, the computational cost can be kept low. Our result indicates that the method can provide accurate and efficient solutions for acoustic wave propagation using adaptive distribution of the calculation points.