62 research outputs found
Some Notes on Summation by Parts Time Integration Methods
Some properties of numerical time integration methods using summation by
parts operators and simultaneous approximation terms are studied. These schemes
can be interpreted as implicit Runge-Kutta methods with desirable stability
properties such as -, -, -, and algebraic stability. Here, insights
into the necessity of certain assumptions, relations to known Runge-Kutta
methods, and stability properties are provided by new proofs and
counterexamples. In particular, it is proved that a) a technical assumption is
necessary since it is not fulfilled by every SBP scheme, b) not every
Runge-Kutta scheme having the stability properties of SBP schemes is given in
this way, c) the classical collocation methods on Radau and Lobatto nodes are
SBP schemes, and d) nearly no SBP scheme is strong stability preserving
Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach
In this paper, we consider Burgers' equation with uncertain boundary and
initial conditions. The polynomial chaos (PC) approach yields a hyperbolic
system of deterministic equations, which can be solved by several numerical
methods. Here, we apply the correction procedure via reconstruction (CPR) using
summation-by-parts operators. We focus especially on stability, which is proven
for CPR methods and the systems arising from the PC approach. Due to the usage
of split-forms, the major challenge is to construct entropy stable numerical
fluxes. For the first time, such numerical fluxes are constructed for all
systems resulting from the PC approach for Burgers' equation. In numerical
tests, we verify our results and show also the advantage of the given ansatz
using CPR methods. Moreover, one of the simulations, i.e. Burgers' equation
equipped with an initial shock, demonstrates quite fascinating observations.
The behaviour of the numerical solutions from several methods (finite volume,
finite difference, CPR) differ significantly from each other. Through careful
investigations, we conclude that the reason for this is the high sensitivity of
the system to varying dissipation. Furthermore, it should be stressed that the
system is not strictly hyperbolic with genuinely nonlinear or linearly
degenerate fields
Stability of step size control based on a posteriori error estimates
A posteriori error estimates based on residuals can be used for reliable
error control of numerical methods. Here, we consider them in the context of
ordinary differential equations and Runge-Kutta methods. In particular, we take
the approach of Dedner & Giesselmann (2016) and investigate it when used to
select the time step size. We focus on step size control stability when
combined with explicit Runge-Kutta methods and demonstrate that a standard I
controller is unstable while more advanced PI and PID controllers can be
designed to be stable. We compare the stability properties of residual-based
estimators and classical error estimators based on an embedded Runge-Kutta
method both analytically and in numerical experiments
Modeling still matters: a surprising instance of catastrophic floating point errors in mathematical biology and numerical methods for ODEs
We guide the reader on a journey through mathematical modeling and numerical
analysis, emphasizing the crucial interplay of both disciplines. Targeting
undergraduate students with basic knowledge in dynamical systems and numerical
methods for ordinary differential equations, we explore a model from
mathematical biology where numerical methods fail badly due to catastrophic
floating point errors. We analyze the reasons for this behavior by studying the
steady states of the model and use the theory of invariants to develop an
alternative model that is suited for numerical simulations. Our story intends
to motivate combining analytical and numerical knowledge, even in cases where
the world looks fine at first sight. We have set up an online repository
containing an interactive notebook with all numerical experiments to make this
study fully reproducible and useful for classroom teaching.Comment: 17 pages, 10 figure
SBP operators for CPR methods: Master's thesis
Summation-by-parts (SBP) operators have been used in the finite difference framework, providing means to prove conservation and discrete stability by the energy method, predominantly for linear (or linearised) equations. Recently, there have been some approaches to generalise the notion of SBP operators and to apply these ideas to other methods. The correction procedure via reconstruction (CPR), also known as flux reconstruction (FR) or lifting collocation penalty (LCP), is a unifying framework of high order methods for conservation laws, recovering some discontinuous Galerkin, spectral difference and spectral volume methods. Using a reformulation of CPR methods relying on SBP operators and simultaneous approximation terms (SATs), conservation and stability are investigated, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015). Extensions of SBP methods with diagonal-norm operators to Burgers’ equation are possible by a skew-symmetric form and the introduction of additional correction terms. An analytical setting allowing a generalised notion of SBP methods including modal bases is described and applied to Burgers’ equation, resulting in an extension of the previously mentioned skew-symmetric form. Finally, an extension of the results to multiple space dimensions is presented
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