A Meshless Volume Scheme

Development of a new meshless technique is described, called the meshless volume scheme. The method is completely meshless, yet retains the low storage and simple flux distribution technique characteristic of finite volume methods. The new method is based on the well-known Taylor series expansion method with least squares. A least squares weighting scheme is introduced that significantly reduces storage and computational requirements compared to other meshless schemes. The scheme is shown to be analogous to finite volume schemes in many ways, but still lacks a discrete telescoping property and is not discretely conservative. The method is applied to the Euler equations in two dimensions. Results are presented which agree well with established methods. I

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An Edge-Averaged Semi-meshless Framework for Numerical Solution of Conservation Laws

Despite the continuous improvement of technology, generation of high-quality meshes around complicated geometry remains a time consuming and even laborious task. In this paper, we propose a semi-meshless computational framework designed to reduce the difficulty in generating the necessary point connectivity for numerically solving conservation laws. The new framework is based on an analytic formulation of conservation laws in a generalized coordinate frame, in which the time derivatives of the conservative variables can be computed using a weighted average of estimates from local stencils. This formulation naturally implies a merit function for generating stencils based on point distances and the orthogonality of the generalized coordinates in the expected stencils. We applied the framework to solving the Euler equations for inviscid flow. Test cases involving airfoils showed that the new computational framework produced numerical solutions with quality similar to their finite volume counterparts. I