166 lines
2.9 KiB
Plaintext
166 lines
2.9 KiB
Plaintext
#LyX 2.1 created this file. For more info see http://www.lyx.org/
|
|
\lyxformat 474
|
|
\begin_document
|
|
\begin_header
|
|
\textclass article
|
|
\use_default_options true
|
|
\maintain_unincluded_children false
|
|
\language english
|
|
\language_package default
|
|
\inputencoding auto
|
|
\fontencoding global
|
|
\font_roman default
|
|
\font_sans default
|
|
\font_typewriter default
|
|
\font_math auto
|
|
\font_default_family default
|
|
\use_non_tex_fonts false
|
|
\font_sc false
|
|
\font_osf false
|
|
\font_sf_scale 100
|
|
\font_tt_scale 100
|
|
\graphics default
|
|
\default_output_format default
|
|
\output_sync 0
|
|
\bibtex_command default
|
|
\index_command default
|
|
\paperfontsize default
|
|
\use_hyperref false
|
|
\papersize default
|
|
\use_geometry false
|
|
\use_package amsmath 1
|
|
\use_package amssymb 1
|
|
\use_package cancel 1
|
|
\use_package esint 1
|
|
\use_package mathdots 1
|
|
\use_package mathtools 1
|
|
\use_package mhchem 1
|
|
\use_package stackrel 1
|
|
\use_package stmaryrd 1
|
|
\use_package undertilde 1
|
|
\cite_engine basic
|
|
\cite_engine_type default
|
|
\biblio_style plain
|
|
\use_bibtopic false
|
|
\use_indices false
|
|
\paperorientation portrait
|
|
\suppress_date false
|
|
\justification true
|
|
\use_refstyle 1
|
|
\index Index
|
|
\shortcut idx
|
|
\color #008000
|
|
\end_index
|
|
\secnumdepth 3
|
|
\tocdepth 3
|
|
\paragraph_separation indent
|
|
\paragraph_indentation default
|
|
\quotes_language english
|
|
\papercolumns 1
|
|
\papersides 1
|
|
\paperpagestyle default
|
|
\tracking_changes false
|
|
\output_changes false
|
|
\html_math_output 0
|
|
\html_css_as_file 0
|
|
\html_be_strict false
|
|
\end_header
|
|
|
|
\begin_body
|
|
|
|
\begin_layout Title
|
|
Boundary layer flow
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\frac{\partial u}{\partial t}-\frac{1}{s^{2}}\frac{\partial^{2}u}{\partial y^{2}}=K(t)
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula $y=0:u=0$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $y=1,\frac{\partial u}{\partial y}=0$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Discretization, FTCD:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}-\frac{1}{s^{2}}\frac{u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}}{\Delta y^{2}}=K^{n}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Rewriting:
|
|
\begin_inset Note Note
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $u_{i}^{n+1}-u_{i}^{n}-\frac{\Delta t}{\Delta y^{2}s^{2}}\frac{u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}}{}=K^{n}\Delta t$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $u_{i}^{n+1}=\Delta tK^{n}+u_{i}^{n}+\frac{\Delta t}{\Delta y^{2}s^{2}}\left(u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}\right)$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $u_{0}=0$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
and
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Formula $u_{N}-u_{n-1}=0$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\[
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_body
|
|
\end_document
|