For a given nonnegative integer α, a matrix A n of size n is called α-Toeplitz if its entries obey the rule A n = [a r−αs ] n−1 r,s=0 . Analogously, a matrix A n again of size n is called α-circulant if A n = a (r−αs) mod n n−1 r,s=0 . Such kind of matrices arises in wavelet analysis, subdivision algorithms and more generally when dealing with multigrid/multilevel methods for structured matrices and approximations of boundary value problems. In this paper we study the singular values of α-circulants and we provide an asymptotic analysis of the distribution results for the singular values of α-Toeplitz sequences in the case where {a k } can be interpreted as the sequence of Fourier coefficients of an integrable function f over the domain (−π, π). Some generalizations to the block, multilevel case, amounting to choose f multivariate and matrix valued, are briefly considered.Theorem 2.1. Let B be an arbitrary (complex) m × n matrix. Then: (a) There exists a unitary m × m matrix U and a unitary n × n matrix V such that U * BV = Σ is an m × n "diagonal matrix" of the following form: Σ = D 0 0 0 , D := diag(σ 1 , . . . , σ r ), σ 1 ≥ σ 2 ≥ · · · ≥ σ r > 0.Here σ 1 , . . . , σ r are the nonvanishing singular values of B, and r is the rank of B.

Summary This work deals with the numerical solutions of two‐dimensional viscous coupled Burgers' equations with appropriate initial and boundary conditions using a three‐level explicit time‐split MacCormack approach. In this technique, the differential operators split the two‐dimensional problem into two pieces so that the two‐step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second‐order accurate in time and fourth‐order convergent in space, whereas it is second‐order convergent in both time and space for high Reynolds numbers problems. This observation shows the utility and efficiency of the considered method compared with a broad range of numerical schemes widely studied in the literature for solving the two‐dimensional time‐dependent nonlinear coupled Burgers' equations. A large set of numerical examples that confirm the theoretical results are presented and critically discussed.

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In this paper, we analyze the three-level explicit time-split MacCormack procedure in the numerical solutions of two-dimensional viscous coupled Burgers' equations subject to initial and boundary conditions. The differential operators split the two-dimensional problem into two pieces so that the two-step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second order accurate in time and fourth convergent in space, while it is second order convergent in both time and space for high Reynolds numbers problems. This shows the efficiency and effectiveness of the considered method compared to a large set of numerical schemes widely studied in the literature for solving the two-dimensional time dependent nonlinear coupled Burgers' equations. Numerical examples which confirm the theoretical results are presented and discussed.Keywords: two-dimensional unsteady nonlinear coupled Burgers' equations, one-dimensional difference operators, two-step MacCormack scheme, three-level explicit time-split MacCormack method, stability and convergence rate.