A new split-domain harmonic balance approach is presented. The split-domain approach combines the conventional multidomain harmonic balance approach with a split-operator technique in a unique way to solve periodic unsteady ow problems ef ciently. The new technique is applied to Burger's equation to obtain solutions for two large-amplitude periodic boundary conditions-a single-frequency sine wave and a simulated wake function. Solutions containing strong moving discontinuities are obtained with Fourier series containing up to 48 frequencies for various grid densities. The split-domain harmonic balance solutions are compared with conventional time-accurate solutions. The differences between the two are found to be asymptotic with respect to the number of Fourier frequencies included. In addition, the harmonic balance approach was found to be sensitive to grid density.
Nomenclaturea = Fourier cosine coef cient Q a = amplitude of unsteady input disturbance b = Fourier sine coef cient Q F = discrete Fourier transform operator O F = harmonic balance ux term vector f = frequency of unsteady input disturbance (1/s) O S = harmonic balance source term vector U = vector of time-sampled dependent variables O U = vector of Fourier coef cients u = scalar dependent variable in Burger's equation°= component of frequency domain ux vector 1¿ = pseudotime numerical integration step size ! = frequency of unsteady input disturbance (rad/s) Subscripts i = grid point n = Fourier frequency number t = differentiation with respect to time x = differentiation with respect to space ¿ = differentiation with respect to pseudotime Superscripts n = time level T = vector transpose ¡1 = operator inverse
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