A shock wave is a form of a propagating disturbance. Like an ordinary wave, a shock carries energy and can propagate through a medium. One of the major difficulties in computational fluid dynamics (CFD) is the generation of suitable meshes for this type of flow. A proper mesh can require considerable effort to generate for a compressible flow problem involving complex configurations. In order to reduce the high cost of mesh generation, various numerical schemes have been proposed. Among them, the relatively recent meshless method employs a very simple technique to generate node points within the problem domain. Existing conventional computational techniques require a mesh to discretize the domain and yield solutions. However, the accuracy of the numerical approach depends upon the quality of the mesh. In this study, a localized meshless method employing multiquadrics (MQs) radial basis functions (RBFs) [1] is used to approximate the terms within the governing equations. The concern of an ill-conditioned matrix, as well as the computational burden of having to evaluate several large matrices, is mitigated by using a localized meshless approach [2].
The localized RBF-MQ technique is used to develop a meshfree Euler solver for inviscid compressible flow. The issue in solving hyperbolic PDEs such as Euler equations is to use a suitable discretization method, which not only can accurately approximate the smooth regions of flow but also have the ability of capturing the discontinuities, or shocks, within the flow field [3]. In the present scheme, a hybrid upwind technique is used to accurately resolve the shock location and propagation.