Several recently published studies regarding flow problems propose schemes of high order of accuracy designed as evolution of traditional methods. A drawback common to these new schemes is the necessity to adopt uniform mesh refinement for solving sharp problems, by increasing the computational cost. Even the so called essentially non-oscillatory and weight essentially non-oscillatory methods suffer of the same drawback and are not suitable to cope with h-adaptive methods due to their definition on finite volumes necessarily of equal diameter. Therefore, in order to overcome the above drawback, the formulation of dynamically locally self h-adaptive processes is designed to achieve the dual purpose to increase the accuracy and to keep as small as possible the number of finite volumes. To define a locally h-adaptive finite volume (FV) scheme need two simple but important tools, namely a particular FV named Bridge FV positioned between two adjacent subdomains and the definition of suitable profiles approximating the fluxes on the FV faces. In this article a new FV method for the numerical solution of convective-diffusive 1D problems is developed. It is conservative, second order in time and space for equal FV, and allows the partitioning of the domain by equal or unequal finite volumes, thus dynamically locally self h-adaptive. The definition of the monotonic profiles is accomplished by means of cubic weighted ν-splines and Taylor expansions. The profile analysis respect to the numerical properties is conducted in the normalized plane with the velocity varying in time and space and gives the flux value on the FV faces. Moreover the flux is assigned by Upwind or by second order back-ward Characteristics if the estimated flux is outside of the unit square or the transformation into the normalized plane is not possible, respectively. The initial-boundary stability and convergence properties of the new method are examined in detail, also in presence of h-adaptivity. In addition, a generalization of the new scheme to 2D and 3D problems is presented. Finally, some numerical test are carried out to verify the properties of the new method, including two CFD problems.
Pennati et al. (Fri,) studied this question.