%?Simulating?the?1-D?Diffusion?equation?（Fourier‘s?equation）?by?the
...Finite?Difference?Method（a?time?march）
%?Numerical?scheme?used?is?a?first?order?upwind?in?time?and?a?second?order
...central?difference?in?space?（both?Implicit?and?Explicit）

%%
%Specifying?Parameters
nx=50;???????????????%Number?of?steps?in?space（x）
nt=30;???????????????%Number?of?time?steps?
dt=0.1;??????????????%Width?of?each?time?step
dx=2/（nx-1）;?????????%Width?of?space?step
x=0:dx:2;????????????%Range?of?x?（02）?and?specifying?the?grid?points
u=zeros（nx1）;???????%Preallocating?u
un=zeros（nx1）;??????%Preallocating?un
vis=0.01;????????????%Diffusion?coefficient/viscosity
beta=vis*dt/（dx*dx）;?%Stability?criterion?（0<=beta<=0.5?for?explicit）
UL=1;????????????????%Left?Dirichlet?B.C
UR=1;????????????????%Right?Dirichlet?B.C
UnL=1;???????????????%Left?Neumann?B.C?（du/dn=UnL）?
UnR=1;???????????????%Right?Neumann?B.C?（du/dn=UnR）?

%%
%Initial?Conditions:?A?square?wave
for?i=1:nx
????if?（（0.75<=x（i））&&（x（i）<=1.25））
????????u（i）=2;
????else
????????u（i）=1;
????end
end

%%
%B.C?vector
bc=zeros（nx-21）;
bc（1）=vis*dt*UL/dx^2;?bc（nx-2）=vis*dt*UR/dx^2;??%Dirichlet?B.Cs
%bc（1）=-UnL*vis*dt/dx;?bc（nx-2）=UnR*vis*dt/dx;??%Neumann?B.Cs
%Calculating?the?coefficient?matrix?for?the?implicit?scheme
E=sparse（2:nx-21:nx-31nx-2nx-2）;
A=E+E‘-2*speye（nx-2）;????????%Dirichlet?B.Cs
%A（11）=-1;?A（nx-2nx-2）=-1;?%Neumann?B.Cs
D=speye（nx-2）-（vis*dt/dx^2）*A;

%%
%Calculating?the?velocity?profile?for?each?time?step
i=2:nx-1;
for?it=0:nt
????un=u;
????disp（u）
????h=plot（xu）;???????%plotting?the?velocity?profile
??
????axis（[0?2?0?3]）
????title（{[‘1-D?Diffusion?with?\nu?=‘num2str（vis）‘?and?\beta?=?‘num2str（beta）];[‘time（\itt）?=?‘num2str（dt*it）]}）
????xlabel（‘Spatial?co-ordinate?（x）?\rightarrow‘）
????ylabel（‘Transport?property?profile?（u）?\rightarrow‘）
????drawnow;?
????refreshdata（h）
????%Uncomment?as?necessary
????%-------------------
????%Implicit?solution
????
????U=un;U（1）=[];U（end）=[];
????U=U+bc;
????U=D\U;
????u=[UL;U;UR];??????????????????????%Dirichlet
????%u=[U（1）-UnL*dx;U;U（end）+UnR*dx];?%Neumann
????%}
????%-------------------
????%Explicit?method?with?F.D?in?time?and?C.D?in?space
????%{
????u（i）=un（i）+（vis*dt*（un（i+1）-2*un（i）+un（i-1））/（dx*dx））;
????%}
end

?屬性????????????大小?????日期????時(shí)間???名稱
-----------?---------??----------?-----??----
?????文件????????2327??2018-10-10?11:06??Diffusion\Diffusion_1D.m
?????文件????????3827??2018-10-10?11:44??Diffusion\Diffusion_2D.m
?????文件????????1110??2018-08-24?10:47??Diffusion\Intro.txt
?????文件????????1334??2018-08-24?10:45??Diffusion\license.txt