Stability and error estimation using entropy functions

Abstract

We have two main objectives in this dissertation: (i) to analyze the splitting mechanism behind entropy stable finite difference approximations of conservative systems; and (ii) to investigate global error estimation for nonlinear, hyperbolic partial differential equations. For symmetrizable systems of conservation laws, Olsson used entropy functions to obtain rigorous stability estimates for a family of finite difference schemes that approximate the original equations [Ols95c]. A key element behind the estimates and the resulting schemes is a splitting process which uses an entropy function to recast the flux derivative into a skew-symmetric form. Gerritsen applied the splitting concept to the compressible Euler equations [Ger96a]. We studied the splitting process through a parameter which defines both the schemes and the family of entropy functions that was used by Gerritsen and Olsson for the Euler equations. Our analysis of this parameter enabled us to compare the schemes' behaviors relative to each other. Our results demonstrate the existence of an optimal value which minimizes errors. Both our theoretical analysis and computational examples show advantages of using the split schemes over their un-split counterparts. These benefits include greater accuracy and efficiency for the solution algorithm, and longer periods of integration. We also illustrate the split schemes' ability to compute the entropy solution when discontinuities (of different types) exist. Our analysis of both the stability estimates and entropy errors provides insights into these behaviors. We also derived an error estimate for the entropy function of symmetrizable hyperbolic systems. Like classical estimates for the computational error, the estimate for the entropy error reflects an accumulation of local truncation errors. We show that the computational and entropy errors converge in a similar fashion. Therefore, the entropy error estimate can be used to monitor and control global accuracy. Since the entropy error estimate utilizes variables already computed by the discretization kernel, it does not add any substantial cost to the solution algorithm. We also demonstrate the feasibility of using the entropy error estimate as a monitor function for local grid adaption purposes.