Methods for the Temporal Approximation of Nonlinear, Nonautonomous Evolution Equations
Eisenmann, Monika LU
(2019)
- Abstract
- Differential equations are an important building block for modeling processes in physics, biology, and social sciences. Usually, their exact solution is not known explicitly though. Therefore, numerical schemes to approximate the solution are of great importance. In this thesis, we consider the temporal approximation of nonlinear, nonautonomous evolution equations on a finite time horizon. We present two independent approaches that can be used to find a temporal approximation of the solution.
As the solution of a nonlinear equation typically lacks global higher-order regularity, it cannot be expected to obtain higher-order convergence rates. Thus, we only concentrate on schemes that are formally of first order.
In the first part of... (More) - Differential equations are an important building block for modeling processes in physics, biology, and social sciences. Usually, their exact solution is not known explicitly though. Therefore, numerical schemes to approximate the solution are of great importance. In this thesis, we consider the temporal approximation of nonlinear, nonautonomous evolution equations on a finite time horizon. We present two independent approaches that can be used to find a temporal approximation of the solution.
As the solution of a nonlinear equation typically lacks global higher-order regularity, it cannot be expected to obtain higher-order convergence rates. Thus, we only concentrate on schemes that are formally of first order.
In the first part of the thesis, we consider the question of how nonsmooth temporal data can be handled. A common method for the approximation of the integral of an irregular function is a Monte Carlo type quadrature rule. We take on this idea and use a similar approach to approximate the solution to a nonautonomous evolution equation. If the data is evaluated at the points of a randomly shifted grid, we can prove the convergence of the backward Euler scheme. Moreover, we prove explicit error estimates. Here, we introduce a
second set of randomized points, where the data is evaluated, and make additional assumptions on the data and the solution.
Secondly, we approximate the solution via an operator splitting based scheme. We work with both an implicit-explicit splitting and a product type splitting. First, we decompose the operator into a monotone and a bounded part. The implicit-explicit splitting is used to obtain one implicit equation that contains the monotone part. The bounded part is solved in an explicit fashion. This way, we only solve as many implicit equations as necessary. Further, we use a product type splitting on the monotone part. Even though this leads to more problems, they are potentially easier to solve individually. For this splitting scheme, we follow a similar approach as in the first part of the thesis. After proving the convergence
of the scheme, we provide error bounds under additional assumptions on both the data and the solution.
In order to provide an interesting field of application, we show that the schemes can be applied for the temporal approximation of certain nonlinear, parabolic problems. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/df9834c4-037c-48a9-a14b-1b2e95f6ca7c
- author
- Eisenmann, Monika
LU
- publishing date
- 2019
- type
- Thesis
- publication status
- published
- subject
- language
- English
- LU publication?
- no
- id
- df9834c4-037c-48a9-a14b-1b2e95f6ca7c
- alternative location
- https://api-depositonce.tu-berlin.de/server/api/core/bitstreams/703950f1-12ff-492b-936b-018f13a3964f/content
- date added to LUP
- 2024-10-07 15:09:55
- date last changed
- 2025-04-04 15:43:35
@phdthesis{df9834c4-037c-48a9-a14b-1b2e95f6ca7c,
abstract = {{Differential equations are an important building block for modeling processes in physics, biology, and social sciences. Usually, their exact solution is not known explicitly though. Therefore, numerical schemes to approximate the solution are of great importance. In this thesis, we consider the temporal approximation of nonlinear, nonautonomous evolution equations on a finite time horizon. We present two independent approaches that can be used to find a temporal approximation of the solution.<br/>As the solution of a nonlinear equation typically lacks global higher-order regularity, it cannot be expected to obtain higher-order convergence rates. Thus, we only concentrate on schemes that are formally of first order.<br/>In the first part of the thesis, we consider the question of how nonsmooth temporal data can be handled. A common method for the approximation of the integral of an irregular function is a Monte Carlo type quadrature rule. We take on this idea and use a similar approach to approximate the solution to a nonautonomous evolution equation. If the data is evaluated at the points of a randomly shifted grid, we can prove the convergence of the backward Euler scheme. Moreover, we prove explicit error estimates. Here, we introduce a<br/>second set of randomized points, where the data is evaluated, and make additional assumptions on the data and the solution.<br/>Secondly, we approximate the solution via an operator splitting based scheme. We work with both an implicit-explicit splitting and a product type splitting. First, we decompose the operator into a monotone and a bounded part. The implicit-explicit splitting is used to obtain one implicit equation that contains the monotone part. The bounded part is solved in an explicit fashion. This way, we only solve as many implicit equations as necessary. Further, we use a product type splitting on the monotone part. Even though this leads to more problems, they are potentially easier to solve individually. For this splitting scheme, we follow a similar approach as in the first part of the thesis. After proving the convergence<br/>of the scheme, we provide error bounds under additional assumptions on both the data and the solution.<br/>In order to provide an interesting field of application, we show that the schemes can be applied for the temporal approximation of certain nonlinear, parabolic problems.}},
author = {{Eisenmann, Monika}},
language = {{eng}},
title = {{Methods for the Temporal Approximation of Nonlinear, Nonautonomous Evolution Equations}},
url = {{https://api-depositonce.tu-berlin.de/server/api/core/bitstreams/703950f1-12ff-492b-936b-018f13a3964f/content}},
year = {{2019}},
}