Moving meshless methods are a new generation of numerical methods for time-dependent PDEs, specially tailored to handle shocks or regions with large gradients. These methods link the moving finite element method (MFE) with meshless methods, where instead of hat functions, bases derived from any meshless methods can be used. An initial distribution of mesh points is found using a suitable monitor function based on the equidistribution principle. A penalty is appended to the energy functional to prevent high mesh velocity and collisions of nodes. Here, we employ a decreasing function of the distance between particles as the viscosity function. We develop a modified version of an algorithm developed earlier, the Moving Element-Free Petrov-Galerkin Viscous Method (MEFPGVM), which combines moving finite element (MFE) and element-free Galerkin (EFG) methods, along with initial mesh equidistribution. Numerical solutions for the 1–D Burgers’ and Korteweg-de Vries equations demonstrate the accuracy of the numerical approximations.

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