A mixed spectral/wavelet method for the solution of the Stokes problem. (English) Zbl 0926.76083
Summary: We present a mixed wavelet/spectral Chebychev method for solving the unsteady two-dimensional Stokes equations in the vorticity-stream function formulation with periodicity condition in one direction. After an appropriate time discretisation of the equations, one has to solve at each time step a stationary Stokes-like problem. A capacitance matrix method is used to eliminate the problem of boundary conditions. This leads to solving a series of Helmholtz problems. The spatial discretisation makes use of the wavelet method in the periodic direction and the spectral collocation Chebychev method in the non-periodic direction. The resolution of the discrete Helmholtz problem is done by means of the diagonalisation technique in the nonperiodic direction. The system then splits into a sequence of one-dimensional periodic Helmholtz problems which are efficiently inverted using FFTs. Numerical tests show both the stability and the accuracy of the method. \(\copyright\) Academic Press.
MSC:
| 76M22 | Spectral methods applied to problems in fluid mechanics |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
fast Fourier transform technique; vorticity-stream function formulation; capacitance matrix method; periodic direction; spectral collocation Chebychev method; non-periodic direction; diagonalisation techniqueReferences:
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