High order accurate finite difference schemes based on symmetry preservation

Construction of invariant compact finite-difference schemes

“…Alternatively, one can use the chain rule to find the extended point transformations. For instance, the transformation expression for the spatial first derivative can be found bersome and not practical for preservation in associated compact finite difference schemes [42]. Hence, for this particular problem, we only choose the subgroups X 1 , X 3 , X 6 , and X 7 for preservation in the associated (non-invariant) compact scheme given in Eq.…”

“…For instance, normalization conditionst (i,n) = 0 andx (i,n) = 0, among infinite possibilities, yield a simple stencil where the symmetry parameters s 6 and s 7 are −t (i,n) and −x (i,n) , respectively. Second, we choose normalization conditions that remove terms from the truncation error of compact schemes under consideration and hence lead to a considerable improvement in numerical accuracy, besides simplifying their numerical representations [42]. In this context, the normalization conditionũ (i,n) x = 0 can be used to determine the symmetry parameter s 1…”

“…( 18) in an arbitrary order. Although it is possible to consider the full Lie algebra and obtain global transformations for the coordinate variables and derivatives, it is sometimes practical to choose only certain subgroups as the form of the point transformations obtained from the full Lie algebra could be cumbersome and not practical for preservation in associated compact finite difference schemes [42]. Hence, for this particular problem, we only choose the subgroups X 1 , X 3 , X 6 , and X 7 for preservation in the associated (non-invariant) compact scheme given in Eq.…”

“…In this paper, we propose a mathematical approach for construction of high order accurate compact finite difference schemes that retain Lie symmetry groups of underlying differential equations. In the pro-posed method that is based on equivariant moving frames, extended symmetry groups of partial differential equations are used to obtain point transformations not only for independent and dependent variables of differential equations, but also for their derivative terms (which is a novel aspect of this paper that was not considered in earlier works [35,36,42]). Once point transformations for derivatives of differential equations are determined, then these transformations are applied to some (non-invariant) base compact finite difference schemes (of a desired order of accuracy) to obtain final invariant (or symmetry preserving) forms of these schemes.…”

“…In the first group [20][21][22][23][24], invariants of difference equations are determined through Lie's infinitesimal approach, and then, a set of these invariants are used to construct invariant schemes that converge to the original differential equations in the continuous limit. In the other group [33][34][35][36][37][38][39][40][41][42][43], point transformations based on symmetry groups of differential equations are applied to some base (non-invariant) numerical schemes, and the unknown symmetry parameters of these transformations are determined through moving frames that are based on Cartan's method of normalization [44].…”