IGG(χ): A new and simple implicit gradient scheme on unstructured meshes

A new and simple implicit Green-Gauss gradient (IGG) scheme for unstructured meshes is proposed exploiting the ideas from the linearity-preserving U-MUSCL scheme to define values at cell faces. We construct an implicit one-parameter family of gradient schemes referred to as IGG(χ) where χ is a free-parameter. The computed gradients are at least first-order accurate on generic polygonal meshes and second-order accurate on uniform Cartesian meshes except when $\chi =1/3$ for which fourth-order accuracy can be realised. A theoretical analysis is carried out to understand the effect of the control parameter χ on accuracy and resolution of the gradients and numerical experiments on various mesh topologies confirm the theoretical findings. Finite volume simulations of the Poisson and Euler equations on Cartesian and unstructured meshes further highlight that the IGG(χ) is a versatile gradient scheme that gives second-order accurate solutions with the iterative convergence of the solver dependent on the choice of the χ parameter. The framework described in this study can also be employed to devise an implicit least-squares gradient scheme that applies equally well to unstructured finite volume and meshfree solvers.