On explicit exponential integrators in the solution of elastic wave propagation equations - Ravelo, Fernando Valdés (2024)
The exponential integrators, a class of numerical methods used to solve differential equations, are the subject of this work. Specifically, we focus on explicit exponential integrators used to solve the differential equations describing the propagation of acoustic and elastic waves, with absorbing boundary conditions, encountered in seismic imaging applications. Among the various methods of exponential integrators, we analyze in detail the Faber polynomial-based method, a generalization of the well-known Chebyshev exponential integrator. Considering the state of the art of the Faber polynomial approximation, we discuss the main limitations of the method and propose solutions for them. Among the theoretical results of the Faber polynomial approximation, we present a more accurate estimate of the approximation error of the method for normal matrices, than the one reported in the literature. We also show the importance of accurate estimates of the operator spectrum to ensure fast convergence of the method. Moreover, based on various numerical experiments, we outline a scheme to obtain eigenvalue estimates using only low-dimensional operators. Among the numerical results, we observe that increasing the degree of Faber polynomials also increases the maximum time step size in temporal integration. Furthermore, in analyzing computational efficiency, we find that using higher degrees of Faber polynomials reduces the number of matrix-vector operations performed. The robustness of our numerical results is ensured by implementing various tests with different levels of complexity. Additionally, we compare the Faber polynomial method with other explicit exponential integrators, such as the Krylov subspace method and high-order Runge-Kutta methods, along with classical low-order methods. Comparisons were made in experimental scenarios simulating real situations encountered in seismic imaging applications. Subsequently, we evaluate the stability, dispersion, numerical convergence, and computational efficiency of these methods. In our analysis, among high-order exponential integrators, the Krylov subspace-based method showed the best convergence results compared to all exponential integration methods. Allowing longer time steps for the same degree of approximation compared to other methods. Notably, when comparing methods for computational efficiency, we observed that high-order numerical methods can achieve efficiency comparable to low-order methods while allowing significantly larger time steps. To highlight other applications that require an efficient solution to the wave equation, we present a new application in the field of mathematical modeling of cancer. As an innovative proposal, we developed a model based on continuum mechanics to simulate the effect of High-Energy Shock Wave (HESW) therapy on the growth of an avascular tumor. In this model, we demonstrate that by adjusting different parameters of the HESW therapy, we can qualitatively reproduce various tumor growth patterns, as reported in the literature. Additionally, we conduct a sensitivity analysis of the model to the various therapy parameters, identifying the most influential elements in tumor growth.