Calculation of eigenvalues of Couette spectral problem by method of regularized traces

One can find the eigenvalues of non-selfadjoint operators only by numerical methods. The use of these methods is associated with large computational difficulties. Therefore, the development of a new method for calculating of eigenvalues of non-self-adjoint operators has great theoretical and practical interest. Non iterative method for finding of eigenvalues of perturbed self-adjoint operators is developed on the basis of the theory of regularized traces. This method is called the method of regularized traces. The linear formulas for computing of the eigenvalues of the discrete operators, which are semi-bounded from below, were found. Using them, one can compute the eigenvalues of perturbed self-adjoint operator with any their number. Note that for this computation it does not matter whether the eigenvalues with less number are known or not. Numerical calculations of eigenvalues for the spectral problems, which are generated by the equations of mathematical physics, show that for large numbers of eigenvalues the proposed formulas give more exact result than the Galerkin method. In addition, the obtained formulas allow to compute the eigenvalues of perturbed self-adjoint operator with very large number, such that the use of the Galerkin method becomes difficult. The algorithm of application of the method of regularized traces for finding of eigenvalues of the Couette spectral problem of hydrodynamic stability theory is constructed. This problem studies the stability of the flow of a tough liquid between two rotating axisymmetric cylinders to small perturbations of the basic flow. A feature of this problem is the fact that the differential operator is a matrix one. Numerical experiments have shown the high computational efficiency of the proposed algorithm of computing of the eigenvalues of the studied spectral problem.
The algorithm of application of the method of regularized traces for spectral problems, which are generated by the matrix discrete operators limited from below, is constructed in the paper.