A Spectrum Enveloping Technique for Iterative Solution of Central Difference Approximations of Convection-Diffusion Equations
Murli M. Gupta · SIAM Journal on Algebraic and Discrete Methods · 1986
When a convection-diffusion equation is discretized using the central difference scheme the resulting coefficient matrix is not diagonally dominant whenever the convection terms are large. If this system of linear equations is solved using the conventional iteration methods, the iterations often fail to converge as some of the eigenvalues of the iteration matrix lie outside the unit circle $C = \{ z: | z |\leqq 1 \}$ in the complex plane. The eigenvalue spectrum of some of the iteration matrices lies inside the infinite strip $S = \{ z: | \operatorname{Real} ( z ) | < 1, | \operatorname{Imag} ( z ) | < \infty \}$. An example is that of the method of simultaneous displacements or the Jacobi method. In such cases, it is possible to enclose the eigenvalue spectrum inside an ellipse with major axis on the imaginary axis and minor axis in the real interval $( - 1,1 )$. This ellipse is used to define a convergent iteration. A practical computational algorithm is described to obtain such an iteration scheme. Numerical examples show that the spectrum enveloping technique works well when the original iterations diverge. When the original iterations converge the spectrum enveloping technique can converge even faster.