Error analysis of implicit Runge-Kutta methods for quasilinear hyperbolic evolution equations

We establish error bounds of implicit Runge-Kutta methods for a class of quasilinear hyperbolic evolution equations including certain Maxwell and wave equations. Our assumptions cover algebraically stable and coercive schemes such as Gauß and Radau collocation methods. We work in a refinement of the analytical setting of Kato\u27s well-posedness theory

Nonlinear Evolution Equations: Analysis and Numerics

The qualitative theory of nonlinear evolution equations is an important tool for studying the dynamical behavior of systems in science and technology. A thorough understanding of the complex behavior of such systems requires detailed analytical and numerical investigations of the underlying partial differential equations

Error Analysis of Exponential Integrators for Nonlinear Wave-Type Equations

This thesis is concerned with the time integration of certain classes of nonlinear evolution equations in Hilbert spaces by exponential integrators. We aim to prove error bounds which can be established by including only quantities given by a wellposedness result. In the first part, we consider semilinear wave equations and introduce a class of first- and second-order exponential schemes. A standard error analysis is not possible due to the lack of regularity. We have to employ appropriate filter functions as well as the integration by parts and summation by parts formulas in order to obtain optimal error bounds. In the second part, we propose two exponential integrators of first and second order applied to a class of quasilinear wave-type equations. By a detailed investigation of the differentiability of the right-hand side we derive error bounds in different norms. In the framework we can treat quasilinear Maxwell’s equations in full space and on a smooth domain as well as a class of quasilinear wave equations. In both parts, we include numerical examples to confirm our theoretical findings

Error analysis of implicit Euler methods for quasilinear hyperbolic evolution systems

In this paper we study the convergence of the semi-implicit and the implicit Euler methods for the time integration of abstract, quasilinear hyperbolic evolution equations. The analytical framework considered here includes certain quasilinear Maxwell\u27s and wave equations as special cases. Our analysis shows that the Euler approximations are well-posed and convergent of order one. The techniques will be the basis for the future investigation of higher order time integration methods and full discretizations of certain quasilinear hyperbolic problems

Exponential integrators for quasilinear wave-type equations

In this paper we propose two exponential integrators of first and second order applied to a class of quasilinear wave-type equations. The analytical framework is an extension of the classical Kato framework and covers quasilinear Maxwell’s equations in full space and on a smooth domain as well as a class of quasilinear wave equations. In contrast to earlier works, we do not assume regularity of the solution but only on the data. From this we deduce a well-posedness result upon which we base our error analysis. We include numerical examples to confirm our theoretical findings

Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation

We consider the development of high order space and time numerical methods based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. As a consequence the nature of the asymptotic limit changes completely, passing from a hyperbolic to a parabolic system. From the computational point of view, standard numerical methods designed for the fluid-dynamic scaling of hyperbolic systems with relaxation present several drawbacks and typically lose efficiency in describing the parabolic limit regime. In this work, in the context of Implicit-Explicit linear multistep methods we construct high order space-time discretizations which are able to handle all the different scales and to capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. Several numerical examples confirm the theoretical analysis