High order methods for hyperbolic PDEs with singular source term
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In this research we consider hyperbolic partial differential equations with singular source term. First we consider the Zerilli equation, which models the phenomenon, when a star or other celestial object colloids with a black hole. In this model, there is no angular momentum. We develop the spectral-finite difference hybrid method which solves the equation very efficiently and accurately yields the quasi-normal modes and the power-law decay profile. This method is very fast compared to other methods. We also consider the sine-Gordon and nonlinear Schrödinger equations with a point-like singular source term. The soliton interaction with such a singular potential yields a critical solution behavior. That is, for the given value of the potential strength or the soliton amplitude, there exists a critical velocity of the initial soliton solution, around which the solution is either trapped by or transmitted through the potential. In this research, we propose an efficient method for finding such a critical velocity by using the generalized polynomial chaos (gPC) method. For the proposed method, we assume that the soliton velocity is a random variable and expand the solution in the random space using the orthogonal polynomials. We consider the Legendre and Hermite chaos with both the Galerkin and collocation formulations. The proposed method finds the critical velocity accurately with spectral convergence. Thus the computational complexity is much reduced. The very core of the proposed method lies in using the mean solution instead of reconstructing the solution. The mean solution converges exponentially while the gPC reconstruction may fail to converge to the right solution due to the Gibbs phenomenon in the random space. Numerical results confirm the accuracy and spectral convergence of the method. For the last problem a hybrid method based on the spectral method and weighted essentially non-oscillatory (WENO) finite difference method is proposed to solve the unsteady transonic equations.