function ForwardandBackwardEulerExplorer()
% GUI for comparing the difference between Forward Euler, Backward Euler
% and Crank Nicolson for a basic time integration problem:
%
% du/dt = lambda*u
%
% The idea behind the GUI is to facilitate the explanation of stability
% regions. Of particular interest is when lambda*dt = 2
% Copyright 2013 The MathWorks, Inc.
handles.FontSize = 12;
handles.Color = [
0.00 0.00 1.00
0.00 0.50 0.00
1.00 0.00 0.00
0.00 0.75 0.75
0.75 0.00 0.75
];
handles.StabilityColor = [
0.729412 0.831373 0.956863
0.7569 0.8667 0.7765
0.9255 0.8392 0.8392
0.7255 0.9255 0.9216
0.9882 0.7569 0.9882
];
handles.LineWidth = 2;
handles.MarkerSize = 10;
handles.StabilityXdom = [-3 3];
handles.StabilityYdom = [-3 3];
%% build the gui
fh = figure('Visible','on','Name','ForwardandBackwardEulerExplorer');
set(fh,'MenuBar','none')
pos_orig = get(fh,'Position');
Width = 750; Height = 750;
set(fh,'Position',[pos_orig(1), pos_orig(2)+pos_orig(4)-Height, Width, Height]);
set(fh,'units','normalized');
handles.textaxis = axes('parent',fh,...
'units','normalized',...
'position',[0 0 1 1],...
'visible','off');
handles.PlotAxes = axes('Parent',fh,...
'Position',[.1, .55, .8, .425]);
xlabel('t','FontSize',handles.FontSize)
ylabel('u','FontSize',handles.FontSize)
set(handles.PlotAxes,'XLim',[0 20]);
hEval = line;
%set(hEval,'xdata',[],'ydata',[]);
set(hEval,'Marker','o','LineStyle','none');%,'Visible','off')
set(hEval,'LineStyle','none');
set(hEval,'MarkerSize',handles.MarkerSize);
set(hEval,'Color',handles.Color(1,:));
set(hEval,'LineWidth',handles.LineWidth);
hExact = line;
%set(hExact,'xdata',[],'ydata',[]);
set(hExact,'LineStyle','--')
set(hExact,'Color',handles.Color(2,:));
set(hExact,'LineWidth',handles.LineWidth);
% store the handles for later use
handles.hExact = hExact;
handles.hEval = hEval;
handles.Legend = legend('Numeric','Exact');
handles.StabilityAxes = axes('Parent',fh,...
'Position',[.1, .075, .4, .425]);
xlabel('Re(\lambda\Delta t)','FontSize',handles.FontSize)
ylabel('Im(\lambda\Delta t)','FontSize',handles.FontSize)
xlim([-3 3]);
ylim([-3 3]);
grid on;
handles.MethodSelector = uicontrol(fh,'Style','popupmenu',...
'String',{'Forward Euler', 'Backward Euler', 'Crank Nicolson'},...
'Value',1,...
'FontSize',handles.FontSize,...
'units','normalized',...
'Position',[.6 .4 .3 .1],...
'CallBack',@UpdatePlot);
handles.LambdaEdit = uicontrol(fh,'Style','edit',...
'String','-2',...
'FontSize',handles.FontSize,...
'units','normalized',...
'Position',[.615 .2 .075 .075],...
'CallBack',@UpdateLambda);
handles.DeltaTEdit = uicontrol(fh,'Style','edit',...
'String','0.5',...
'FontSize',handles.FontSize,...
'units','normalized',...
'Position',[.615 .1 .075 .075],...
'CallBack',@UpdateDeltaT);
handles.WhyButton = uicontrol(fh,'Style','pushbutton',...
'String','Why',...
'FontSize',handles.FontSize,...
'units','normalized',...
'Position',[.7 .02 .2 .05],...
'CallBack',@WhyButtonPress);
% text
axes(handles.textaxis)
str = {
'v^{n+1} = v^{n} + \lambda\Delta t v^{n}';
'v^{n+1} = (1 + \lambda\Delta t) v^{n}';
};
handles.MethodText = text(.75,.4,str,...
'unit','normalized',...
'HorizontalAlignment','center',...
'FontSize',handles.FontSize);
handles.LambdaText = text(.6,.24,'\lambda:',...
'unit','normalized',...
'HorizontalAlignment','right',...
'FontSize',handles.FontSize);
handles.DeltaTText = text(.6,.14,'\Delta t:',...
'unit','normalized',...
'HorizontalAlignment','right',...
'FontSize',handles.FontSize);
% add sliders
handles.LambdaSlider = uicontrol(fh,'Style','slider',...
'Min',-10,'Max',10,...
'Value',-2,...
'FontSize',handles.FontSize,...
'units','normalized',...
'Position',[.715,0.2, .25, .075],...
'CallBack',@UpdateLambda);
handles.DeltaTSlider = uicontrol(fh,'Style','slider',...
'Min',0.1,'Max',10,...
'Value',0.5,...
'FontSize',handles.FontSize,...
'units','normalized',...
'Position',[.715,0.1, .25, .075],...
'CallBack',@UpdateDeltaT);
setappdata(fh,'handles',handles)
UpdateLambda(handles.LambdaEdit);
end
function UpdateLambda(hObject, eventdata)
fh = get(hObject,'Parent');
handles = getappdata(fh,'handles');
style = get(hObject,'Style');
switch lower(style)
case 'edit'
lambda = str2double(get(handles.LambdaEdit,'String'));
case 'slider'
lambda = get(handles.LambdaSlider,'Value');
otherwise
error('Unexpected style');
end
lambdamin = get(handles.LambdaSlider,'Min');
lambdamax = get(handles.LambdaSlider,'Max');
if (lambda<lambdamin)
lambda = lambdamin;
end
if (lambda>lambdamax)
lambda = lambdamax;
end
% set the lambdas
set(handles.LambdaEdit, 'String',sprintf('%.1f',lambda));
set(handles.LambdaSlider,'Value', lambda);
UpdatePlot(hObject);
end
function UpdateDeltaT(hObject, eventdata)
fh = get(hObject,'Parent');
handles = getappdata(fh,'handles');
style = get(hObject,'Style');
switch lower(style)
case 'edit'
dt = str2double(get(handles.DeltaTEdit,'String'));
case 'slider'
dt = get(handles.DeltaTSlider,'Value');
otherwise
error('Unexpected style');
end
dtmin = get(handles.DeltaTSlider,'Min');
dtmax = get(handles.DeltaTSlider,'Max');
if (dt<dtmin)
dt = dtmin;
end
if (dt>dtmax)
dt = dtmax;
end
% set the lambdas
set(handles.DeltaTEdit, 'String',sprintf('%.1f',dt));
set(handles.DeltaTSlider,'Value', dt);
UpdatePlot(hObject);
end
function UpdatePlot(hObject, eventdata)
fh = get(hObject,'Parent');
handles = getappdata(fh,'handles');
% get relavent handles
hExact = handles.hExact;
hEval = handles.hEval;
lambda = str2double(get(handles.LambdaEdit,'String'));
dt = str2double(get(handles.DeltaTEdit,'String'));
method = get(handles.MethodSelector,'Value');
Color = handles.Color;
Scolor = handles.StabilityColor;
LW = handles.LineWidth;
xdom = handles.StabilityXdom;
ydom = handles.StabilityYdom;
% remove current children
current_children = get(handles.StabilityAxes,'Children');
delete(current_children);
switch method
case 1
g = 1+lambda*dt;
str = {
'v^{n+1} = v^{n} + \Delta t f(v^{n},t^{n})';
'v^{n+1} = v^{n} + \lambda\Delta t v^{n}';
'v^{n+1} = (1 + \lambda\Delta t) v^{n}';
};
type = 'ef';
case 2
g = 1/(1-lambda*dt);
str = {
'v^{n+1} = v^{n} + \Delta t f(v^{n+1},t^{n+1})';
'v^{n+1} = v^{n} + \lambda\Delta t v^{n+1}'
'v^{n+1} = 1/(1-\lambda\Delta t) v^{n}'
};
type = 'eb';
case 3
g = (1+lambda*dt/2)/(1-lambda*dt/2);
str = {
'v^{n+1} = v^{n} + \Delta t 0.5( f(v^{n+1},t^{n+1})+f(v^{n},t^{n}) )';
'v^{n+1} = v^{n} + \lambda\Delta t 0.5 (v^{n+1}+v^{n})'
'v^{n+1} = (1+0.5\lambda\Delta t)/(1-0.5\lambda\Delta t) v^{n}'
};
type = 'cn';
otherwise
error();
end
N = ceil(20/dt);
t = dt*[0:N];
v = g.^[0:N];
te = linspace(0,20,100);
ue = exp(lambda*te);
set(hExact,'xdata',te,'ydata',ue);
set(hEval ,'xdata',t ,'ydata',v);
set(handles.MethodText,'String',str);
% add the stability plots
[hp,hl] = drawstabcontour(type,xdom,ydom,handles.StabilityAxes);
set(hp,'FaceAlpha',0.2);
set(hp,'FaceColor',Scolor(1,:));
set(hp,'LineStyle','none');
set(hl,'color',Color(1,:));
set(hl,'linewidth',3);
% add lambdadeltat dot
axes(handles.StabilityAxes);
h = line(lambda*dt,0);
set(h,'marker','o')
set(h,'MarkerSize',handles.MarkerSize)
set(h,'Color',handles.Color(1,:));
set(h,'LineWidth',handles.LineWidth);
end
function WhyButtonPress(hObject, eventdata)
if (ispc)
web('html\WhyExplicitvsImplicit.html')
else
if (isunix)
web('html/WhyExplicitvsImplicit.html')
else
warning('WarnTests:convertTest','Uknown platfrom ispc=0 & isunix=0, \nrun >> web(''html\\explicitvsimplicit.html'')')
end
end
end
function [hpatch, hline] = drawstabcontour(type,xdom,ydom,axes_handle,color)
if (nargin < 1), type = 'eb'; end;
if (nargin < 2), xdom = [-3,3]; end;
if (nargin < 3), ydom = [-3,3]; end;
if (nargin < 4), axes_handle = gca; end;
if (nargin < 5), color = [0 0 1]; end;
x = linspace(xdom(1),xdom(2),301);
y = linspace(ydom(1),ydom(2),301);
[xx,yy] = meshgrid(x,y);
zz = xx + 1i*yy;
switch type
case {'eb','eulerbackward'}
gamma = eulerbackamp(zz);
case {'ef','eulerforward'}
gamma = eulerforwardamp(zz);
case 'rk2'
gamma = rk2amp(zz);
case 'rk4'
gamma = rk4amp(zz);
case 'irk2'
gamma = irk2amp(zz);
case 'ab2'
gamma = ab2amp(zz);
case 'ab3'
gamma = ab3amp(zz);
case 'am2'
gamma = am2amp(zz);
case {'cn','cranknicolson'}
gamma = cnamp(zz);
case 'bdf2'
gamma = bdf2amp(zz);
case 'bdf3'
gamma = bdf3amp(zz);
otherwise
error('unknown integrator type')
end
gammaabs = abs(gamma);
C = contourc(x,y,gammaabs,[1,1]);
xline = C(1,2:end);
yline = C(2,2:end);
xp = [xdom(1) xdom(2) xdom(2) xdom(1) xdom(1) xline];
yp = [ydom(1) ydom(1) ydom(2) ydom(2) ydom(1) yline];
% need to handle special case of cranknicolson and irk2
switch type
case {'cn', 'cranknicolson', 'irk2'}
xp = [0.0 xdom(2) xdom(2) 0.0 0.0 ];
yp = [ydom(1) ydom(1) ydom(2) ydom(2) ydom(1)];
otherwise
% don't need to do anything
end
axes(axes_handle);
hpatch = patch(xp,yp,color,'FaceAlpha',.2,'LineStyle','none');
hline = line(xline,yline);
set(hline,'color',color,'linewidth',3);
end
function gamma = eulerbackamp(lamt)
gamma = 1./(1-lamt);
end
function gamma = eulerforwardamp(lamt)
gamma = 1+lamt;
end
function gamma = rk2amp(lamt)
v1 = 1;
v2 = 1 + 0.5.*lamt.*v1;
gamma = 1+lamt.*v2;
end
function gamma = rk4amp(lamt)
v1 = 1;
v2 = 1 + 0.5.*lamt.*v1;
v3 = 1 + 0.5.*lamt.*v2;
v4 = 1 + lamt.*v3;
gamma = 1 + lamt.*(1/6*v1 + 1/3*v2 + 1/3*v3 + 1/6*v4);
end
function gamma = irk2amp(lamt)
A = [1/4, 1/4-sqrt(3)/6; 1/4+sqrt(3)/6, 1/4];
b = [1/2, 1/2];
c = [1/2-sqrt(3)/6, 1/2+sqrt(3)/6];
d = 1-lamt.*(A(1,1)+A(2,2))+lamt.^2.*(A(1,1)*A(2,2)-A(2,1)*A(1,2));
v1 = 1./d.*(1-lamt*A(2,2) + lamt*A(1,2));
v2 = 1./d.*(lamt*A(2,1) + 1 - lamt*A(1,1));
gamma = 1 + lamt.*(b(1)*v1 + b(2)*v2);
end
function gamma = cnamp(lamt)
gamma = (1+0.5*lamt)./(1-0.5*lamt);
end
function gamma = ab2amp(lamt)
a = 1;
b = -(1+1.5*lamt);
c = 0.5*lamt;
disc = b.^2 - 4.*a.*c;
r1 = 1/(2*a).*(-b+sqrt(disc));
r2 = 1/(2*a).*(-b-sqrt(disc));
gamma = max(abs(r1),abs(r2));
end
function gamma = ab3amp(lamt)
a = zeros(4,1);
b = zeros(4,1);
a(1) = 1;
a(2) = -1;
b(1) = 0;
b(2) = 23/12;
b(3) = -4/3;
b(4) = 5/12;
gamma = zeros(size(lamt));
for k1 = 1:size(lamt,1)
for k2 = 1:size(lamt,2)
p = a - b*lamt(k1,k2);
R = roots(p);
gamma(k1,k2) = max(abs(R));
end
end
end
function gamma = am2amp(lamt)
a = 1-5/12*lamt;
b = -(1+2/3*lamt);
c = 1/12*lamt;
disc = b.^2 - 4.*a.*c;
r1 = 1./(2.*a).*(-b+sqrt(disc));
r2 = 1./(2.*a).*(-b-sqrt(disc));
gamma = max(abs(r1),abs(r2));
end
function gamma = bdf2amp(lamt)
a = 1-2/3*lamt;
b = -4/3;
c = 1/3;
disc = b.^2 - 4.*a.*c;
r1 = 1./(2.*a).*(-b+sqrt(disc));
r2 = 1./(2.*a).*(-b-sqrt(disc));
gamma = max(abs(r1),abs(r2));
end
function gamma = bdf3amp(lamt)
a = 1-6/11*lamt;
b = -18/11;
c = 9/11;
d = -2/11;
gamma = zeros(size(lamt));
for k1 = 1:size(lamt,1)
for k2 = 1:size(lamt,2)
R = roots([a(k1,k2),b,c,d]);
gamma(k1,k2) = max(abs(R));
end
end
end