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In present paper, a two-dimensional numerical study on a standing-wave thermoacoustic engine was performed with compressible SIMPLE algorithm based on a pressure-correction method. First, the simulation model was developed, and the time-dependent compressible thermoacoustic engine system was chosen through substantive numerical tests. Appropriate governing equations for mass, momentum and energy were introduced. Then, the computational results of the onset of the self-excited oscillations across the entire evolution process and the acoustical characteristics of the pressure and velocity wave were presented and analyzed. In addition, the standing-wave of the pressure and velocity along the center of the two stacks are investigated. The crucial nonlinear phenomenon that cannot be captured by the existing linear theory, like high harmonic frequencies, is also revealed in present paper. It is concluded that compressible SIMPLE algorithm could be employed in our future work to simulate and optimize thermoacoustic system. The present result is an important step toward development to predict the high-amplitude thermoacoustic systems and optimize thermoacoustic engine performance. [ABSTRACT FROM AUTHOR]