Implicit finite volume methods for approximating hyperbolic conservation laws

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2023-12

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In this dissertation, we consider high order accurate, implicit, finite volume, weighted essentially non-oscillatory (WENO) schemes for solving hyperbolic conservation laws. Our schemes are locally mass conservative and reduce oscillation in the solution. We solve the problem using the method of lines, which divides the scheme into space and time approximations. For the space discretization, we use weighted essentially non-oscillatory (WENO) reconstructions with adaptive order (WENO-AO). We give background on the reconstructions and numerically analyze the effects of using high order reconstructions in our time-stepping methods. For the evolution in time, we use implicit methods in order to take time-steps larger than the CFL constraint. The first time-stepping method is a self-adaptive theta scheme. We numerically analyze the implementation of two derivations of the self-adaptive theta scheme. We prove linear L-stability for one of the derivations. We show the self-adaptive theta schemes to be an improvement on well-known low order schemes, namely, backward Euler, by means of flux corrected transport. Second are implicit Runge-Kutta schemes. We generalize the moderated (formally referred to as adaptive) Runge-Kutta method, which combines the third order Radau IIA method with composite backward Euler. The moderated Runge-Kutta method, as well as its generalization, is high order accurate in time but drops in order near discontinuities in the solution. We present a reformulation of implicit Runge-Kutta methods in order to moderated by the self-adaptive theta method, which does not fit the traditional Butcher tableau structure. This new moderated method is comparable, if not an improvement in some applications, to the original. The moderated methods, however, are not guaranteed to satisfy the maximum principle. Thus, we again implement flux corrected transport to ensure physically relevant solutions. Applications to linear and nonlinear problems are discussed. Computational results show that the schemes of interest can handle various test problems well.

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