Konvergenz Finiter Differenzenverfahren für nichtlineare hyperbolischparabolische Systeme
(1993) In Diss. Math. Wiss. ETH Zürich, Nr. 10273 Abstract
 This thesis presents a new technique to prove the convergence of finite difference methods applied to nonlinear Systems arising in computational fluid dynamics.
The underlying Systems are either hyperbolic such as the Euler equations or mixed hyperbolicparabolic like the NavierStokes equations. We analyze implicit finitedifference methods. As the argument is based on the concept of consistency and stability, we obtain convergence results for classical Solutions. An important point is that the results are achieved unconditional to the order of consistency of the scheme. It is known that the construction of highorder methods in computational fluid dynamics is much more difficult than in the case of ordinarydifferential... (More)  This thesis presents a new technique to prove the convergence of finite difference methods applied to nonlinear Systems arising in computational fluid dynamics.
The underlying Systems are either hyperbolic such as the Euler equations or mixed hyperbolicparabolic like the NavierStokes equations. We analyze implicit finitedifference methods. As the argument is based on the concept of consistency and stability, we obtain convergence results for classical Solutions. An important point is that the results are achieved unconditional to the order of consistency of the scheme. It is known that the construction of highorder methods in computational fluid dynamics is much more difficult than in the case of ordinarydifferential equations.
The analysis uses ideas from LöpezMarcos and SanzSerna [16] to establish local stability regions around a pilot function. The introduction of this pilot function is the essential idea leading to success. The pilot function is constructed in such a way that it is highly consistent with the scheme and that it converges to the Solution of the underlying system when the stepsize tends to zero.
Strang made use of a similar idea [22]. But, in contrast to Strang's
concept, we do not use the stability of the scheme linearized at the
classical Solution of the partial differential equation. Instead we linearize the scheme at the pilot function. By means of energy estimates, we show the stability of this linearization. Finally, the convergence of the Solution of the discrete method to the pilot function as well as to the desired classical Solution is proved. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1224390
 author
 Schroll, Achim ^{LU}
 opponent

 Lubich, Christian, Universität Würzburg
 organization
 publishing date
 1993
 type
 Thesis
 publication status
 published
 subject
 in
 Diss. Math. Wiss. ETH Zürich, Nr. 10273
 defense location
 ETHZurich
 defense date
 19930727 10:15
 language
 German
 LU publication?
 yes
 id
 7c7b1bc1590a44acb38d8158755b46cc (old id 1224390)
 date added to LUP
 20160316 18:01:45
 date last changed
 20180529 09:23:46
@phdthesis{7c7b1bc1590a44acb38d8158755b46cc, abstract = {This thesis presents a new technique to prove the convergence of finite difference methods applied to nonlinear Systems arising in computational fluid dynamics.<br/><br> <br/><br> The underlying Systems are either hyperbolic such as the Euler equations or mixed hyperbolicparabolic like the NavierStokes equations. We analyze implicit finitedifference methods. As the argument is based on the concept of consistency and stability, we obtain convergence results for classical Solutions. An important point is that the results are achieved unconditional to the order of consistency of the scheme. It is known that the construction of highorder methods in computational fluid dynamics is much more difficult than in the case of ordinarydifferential equations.<br/><br> <br/><br> The analysis uses ideas from LöpezMarcos and SanzSerna [16] to establish local stability regions around a pilot function. The introduction of this pilot function is the essential idea leading to success. The pilot function is constructed in such a way that it is highly consistent with the scheme and that it converges to the Solution of the underlying system when the stepsize tends to zero.<br/><br> <br/><br> Strang made use of a similar idea [22]. But, in contrast to Strang's<br/><br> concept, we do not use the stability of the scheme linearized at the<br/><br> classical Solution of the partial differential equation. Instead we linearize the scheme at the pilot function. By means of energy estimates, we show the stability of this linearization. Finally, the convergence of the Solution of the discrete method to the pilot function as well as to the desired classical Solution is proved.}, author = {Schroll, Achim}, language = {ger}, school = {Lund University}, series = {Diss. Math. Wiss. ETH Zürich, Nr. 10273}, title = {Konvergenz Finiter Differenzenverfahren für nichtlineare hyperbolischparabolische Systeme}, year = {1993}, }