The operator compact implicit method for parabolic equations

Generalized OCI Schemes for Boundary Layer Problems

“…In particular, as pointed out in, e.g. [5], [6], [12], [13], [19], [25], such schemes when applied to (1.2) on a uniform grid have an inherent formal cell Reynolds number limitation. Namely, with a uniform mesh length h, d = / = 0, and b constant, one finds that the cell Reynolds number bh/e must be bounded by some constant depending on the scheme in order to avoid spurious oscillations or gross inaccuracies.…”

“…This representation for Lu is said to be explicit when qj= qt=> 0, and implicit otherwise. Following the terminology of [6], a scheme of the form (1.3) will be referred to as an operator compact implicit (OCI) scheme if it is a formally fourth-order accurate representation of Lu (i.e., if its appropriately scaled truncation error is 0(h4) for fixed e). Note that formal fourth-order accuracy is the highest that can be obtained by a scheme of the form (1.3) (cf.…”

Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations

“…In a procedure in which all the three formulae are needed, as for the case of an incompressible staggered Navier-Stokes solver, the total cost of the fourth-order set of discretizations can be estimated to be 34N. If one employs the set of compact approximations given by Equations (7), (9) and (11), together with the evaluation of the standard fourth-order Padé scheme Equation (1), the total cost can be estimated to be 27N operations, 10N belonging to the standard Padé scheme Equation (1), 5N for each of the formulae Equations (7) and (9) and 7N for Equation (11). The operation-count saving is even more pronounced if one compares the proposed technique with standard formulae, Equations (30)- (32), or with optimized sixth-order schemes.…”

“…They also obtained formulae for second derivatives and for the case of nonuniform mesh, and applied their schemes to a variety of fluid flow problems. Ciment et al [11] developed the theory of the "Operator Compact Implicit Method", in which new classes of compact schemes were derived for parabolic problems by applying the compact technique to the full spatial convection-diffusion operator. Adam [12], while investigating the problem of boundary condition closures, proposed also a new method for obtaining a second derivative approximation for interior points through the so-called "explicit elimination", in which the second derivative is expressed as a function of the values and of the computed set of first derivatives, in a way which shares some similarities with the technique adopted in this paper.…”

A New Time Stepping Method for Solving One Dimensional Burgers' Equations

“…For example, one may refer to the approaches based on the finite difference method ( [2,12,16,18,20,23,27]), Galerkin method ( [11,17,33]), finite element method ( [1,8,10,19,24,28]), spectral method ( [4,6,26]) and cubic spline and sinc-function methods ( [1,24,31,32] etc. Most of the mentioned numerical approaches are using various computational techniques in spatial discretizations to get more accurate approximations.…”