An Implicit Gradient Meshfree Formulation for Convection-Dominated Problems
Meshfree approximations are ideal for the gradient-type stabilized Petrov–Galerkin methods used for solving Eulerian conservation laws due to their ability to achieve arbitrary smoothness, however, the gradient terms are computationally demanding for meshfree methods. To address this issue, a stabilization technique that avoids high order differentiation of meshfree shape functions is introduced by employing implicit gradients under the reproducing kernel approximation framework. The modification to the standard approximation introduces virtually no additional computational cost, and its implementation is simple. The effectiveness of the proposed method is demonstrated in several benchmark problems.
KeywordsConvection-dominated problems reproducing kernel particle method stabilization implicit gradients
The support of this work by US Army Engineer Research and Development Center under contract W912HZ-07-C-0019 is greatly acknowledged.