Fourth order compact schemes for variable coefficient parabolic problems with mixed derivatives

Abstract

In this article, we have developed a higher order compact numerical method for variable coefficient parabolic problems with mixed derivatives. The finite difference scheme, presented here for two-dimensional domains, is based on fourth order spatial discretization. The time discretization has been carried out using using second order Crank-Nicolson. The present scheme shows good dispersion relation preserving property and has been thoroughly investigated for stability. The discrete Fourier analysis shows that the method is unconditionally stable. The fact that the method has been particularly developed for parabolic equations with mixed derivatives makes it suitable for solving incompressible Navier-Stokes (N-S) equations in irregular domains. To verify the proposed method, several problems with exact and benchmark solutions has been investigated. The proposed compact discretization has been extended to tackle flows of varying complexities governed by the twodimensional unsteady N-S equations in domain beyond rectangular. The results show good agreement for all the problems considered.

A new high-order compact finite difference scheme based on precise integration method for the numerical simulation of parabolic equations

“…In Eqs. (19) and (27), the identity matrix I cannot be directly added to the incremental matrix T o . T o is a tiny matrix (this is a matrix whose elements all approach zero).…”

“…However, these compact finite-difference schemes are not asymptotically stable on uniform grids. The fourth-order exact compact difference scheme for mixed derivative parabolic problems with variable coefficients discussed by Sen [18,19] provides a viable scheme for this paper. Gordin [20] applied the Richardson extrapolation method to improve a fourth-order CFDS to sixth order in 1D parabolic equations and Schrödinger-type equations.…”

A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system

“…In that same year, Kumar [29] discussed a high-order compact difference scheme for singularly perturbed reaction diffusion problems on a new Shish Kin mesh. Sen [30,31] discussed the fourth-order exact compact difference scheme for mixed derivative parabolic problems with variable coefficients.…”

“…In order to verify the effectiveness of the improvement proposed above, we consider the system of two-dimensional Burgers' equations(6) proposed in Sec. 2.2.2, over a square domain [0, 1] × [0, 1], with the initial and boundary conditions(31) and the analytical solutionu(x, y, t) = 2πω ∞ α,β=0 αC αβ exp[−(α 2 +β 2 )π 2 ωt] sin(απx) cos(βπy) ∞ α,β=0 C αβ exp[−(α 2 +β 2 )π 2 ωt] cos(απx) cos(βπy) v(x, y, t) = 2πω ∞ α,β=0 βC αβ exp[−(α 2 +β 2 )π 2 ωt]cos(απx) sin(βπy) ∞ α,β=0C αβ exp[−(α 2 +β 2 )π 2 ωt] cos(απx) cos(βπy) (107) The numerical and analytical solutions of two-dimensional examples are present in Tables 16,17 with Re = 100 and τ = 5 × 10 −5 . The numerical solutions of the different time are presented in Figs.…”

Compact Integration Rules as a quadrature method with some applications

“…These terms cause a numerical dispersion and dissipation in the solution[4,5,6,7]. The use of Fourier analysis for assessing the numerical dispersion and dissipation of schemes, also named as the modified wavenumber approach, is widespread[8,9,10,11]. The numerical behaviour of CIR will be explored next by carrying out a modified wavenumber study.…”