Presented is the HLLG (Harten, Lax and van Leer with Gradient inclusion) method for application to the numerical solution of general Partial Differential Equations (PDEs) in conservation form. The HLLG method is based on the traditional HLL method with formal mathematical inclusion of gradients of conserved properties across the control volume employed for flux derivation. The simple extension demonstrates that conventional higher extensions of the HLL method are mathematically inconsistent and produce various numerical instabilities. The HLLG method, with higher order extensions consistent with the flux derivation, is absent of (or less affected by) the said numerical instabilities. The HLLG method is then applied to solutions of the Euler Equations and the simulation of 1D argon RF plasma simulation.