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dif1d_orth.m
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dif1d_orth.m
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clc
clear all
close all
%SOL=@(x) sin(pi*x)+1;
%MSOL=@(x)-pi^2*sin(pi*x);
SOL=@(x) cos(pi*x)-1;
MSOL=@(x)-pi^2*cos(pi*x);
DIF=1.0;
ALPHA=0.9;
XMIN=0.0;
XMAX=1.0;
N=20;
%KV's
% Immer feiner werdendes Gitter nach Lehrbuch
%X = zeros(1,N+1);
%for I=1:N+1
% X(I) = XMIN + (ALPHA^(I-1)-1)/(ALPHA^N-1)*(XMAX-XMIN);
%end
X=linspace(XMIN,XMAX,N+1);
% N halbieren
%for j=1:1 % jeder Durchlauf entfernt Hälfte der Punkte
% fprintf('abc');
%
%idx =1;
%X2=0;
%for i=1:N+1
% if mod(i, 2)==1
% X2(idx)=X(i);
% idx=idx+1;
% end
%end
%X=X2;
%N=length(X)-1;
%end
XC = (X(1:N)+X(2:N+1))/2;
XCR = [XMIN, XC, XMAX];
%%% ANALYTISCHE LÖSUNG
TA = zeros(1, N);
for I=1:N
TA(I)=SOL(XC(I));
end
%figure(1)
%plot(XC, TA, 'x-');
%title('Analytische Loesung')
%%% FVM Lösung
% Randbedingungen
RBE = SOL(XMAX);
RBW = SOL(XMIN);
% Koeffizienten speichern
AP = zeros(1, N);
AE = zeros(1, N);
AW = zeros(1, N);
for I=1:N
DXE = XCR(I+2)-XCR(I+1);
DXW = XCR(I+1)-XCR(I);
AE(I) = DIF/(DXE);
AW(I) = DIF/(DXW);
AP(I) = -AE(I)-AW(I);
end
% Gesamtgleichungssystem aufstellen
A = zeros(N);
b = zeros(N,1);
for I=1:N
b(I) = MSOL(XC(I))*(X(I+1)-X(I));
% Hauptdiagonale
A(I,I) = AP(I);
% Westliche Nebendiagonale
if mod(I,N)==1
b(I) = b(I)-AW(I)*RBW;
else
A(I, I-1) = AW(I);
end
% Östliche Nebendiagonale
if mod(I,N)==0
b(I) = b(I)-AE(I)*RBE;
else
A(I, I+1) = AE(I);
end
end
T=A\b;
%figure(2)
%plot(XC, T, 'x-');
%title('Numerische Loesung')
%%% Lösungsfehler berechnen
SERR=0.0;
ERR=zeros(1, N);
for I=1:N
ERR(I)=T(I)-TA(I);
SERR=SERR+ERR(I)^2;
end
SERR=sqrt(SERR/(N));
%figure(3)
%plot(XC, ERR, 'x-');
%title('Loesungsfehler')
%fprintf('Summierter Fehler %16.10e N=%g\n', SERR, length(X));
%%%% ORDNUNG BESTIMMEN
% für verdünntes Gitter
% cos, äquidistand
ERR5=3.3333911793e-02;
ERR10=8.2165441917e-03;
ERR20=2.0468563747e-03;
ERR40=5.1125958817e-04;
ERR80=1.2778649795e-04;
fprintf('Summierter Fehler %16.10e N=%g\n', SERR, N);
op=log((ERR5)/(ERR10))/log(2);
fprintf('Ordnung des Verfahrens %16.10e \n',op );
op=log((ERR10)/(ERR20))/log(2);
fprintf('Ordnung des Verfahrens %16.10e \n',op );
op=log((ERR20)/(ERR40))/log(2);
fprintf('Ordnung des Verfahrens %16.10e \n',op );
op=log((ERR40)/(ERR80))/log(2);
fprintf('Ordnung des Verfahrens %16.10e \n',op );
%%%% RESIDUUM BERECHNEN
t=zeros(N, 1);
for I=1:N
t(I)=SOL(XC(I));
end
RES=A*t-b;
figure(4)
plot(XC,RES,'x-');
xlabel('XC')
ylabel('RES')
title('Residuum')
%%%% Truncation Error berechnen
% b ohne Randwerte
TERR=zeros(N, 1);
for I=1:N
b(I) = MSOL(XC(I));
end
% Source term
TERRS=zeros(N,1);
for I=1:N
% benötigte Werte zwischenspeichern
XP=XCR(I+1);
Xe=X(I+1);
Xw=X(I);
DX = Xe-Xw;
fP=b(I);
% west
if I==1
fE=b(I+1);
XE=XCR(I+2);
TERRS(I) = 1/(6*DX)*((fE-fP)/(XE-XP)-(fP-RBW)/(XP-Xw))*((Xe-XP)^3-(Xw-XP)^3);
% east
elseif I==N
fW=b(I-1);
XW=XCR(I);
TERRS(I) = 1/(6*DX)*((RBE-fP)/(Xe-XP)-(fP-fW)/(XP-XW))*((Xe-XP)^3-(Xw-XP)^3);
% central
else
fE=b(I+1);
XE=XCR(I+2);
fW=b(I-1);
XW=XCR(I);
TERRS(I) = 1/(6*DX)*((fE-fP)/(XE-XP)-(fP-fW)/(XP-XW))*((Xe-XP)^3-(Xw-XP)^3);
end
end
% Diffusion Term east
TERRDE=zeros(N,1);
TERRDW=zeros(N,1);
for I=1:N
% benötigte Werte zwischenspeichern
XP=XCR(I+1);
TP=T(I);
fP=b(I);
Xe=X(I+1);
Xw=X(I);
DX = Xe-Xw;
% west
if I==1
XE=XCR(I+2);
TE=T(I+1);
fE=b(I+1);
Xee=X(I+2);
XEE=XCR(I+3);
TEE=T(I+2);
De = (Xe-XP)/(XE-XP);
Te = De * TE + (1-De) * TP;
TERRDE(I) = 1/(2*(XE-XP))*((TEE-TP)/(XEE-XP)-(TE-RBW)/(XE-Xw))...
* (((XP-Xe)^2-(XE-Xe)^2)/(XE-XP))...
+ (1/(Xee-Xe)*((TEE-TE)/(XEE-XE)-(TE-TP)/(XE-XP)) - 1/(Xe-Xw)*((TE-TP)/(XE-XP)-(TP-RBW)/(XP-Xw)))...
* 1/(6*(XE-XP))*(((XP-Xe)^3-(XE-Xe)^3)/(XE-XP));
TERRDW(I) = -1/2*((Te-RBW)/(Xe-Xw) - (TP-RBW)/(XP-Xw));
% west +1
elseif I==2
XE=XCR(I+2);
TE=T(I+1);
fE=b(I+1);
Xee=X(I+2);
XEE=XCR(I+3);
TEE=T(I+2);
XW=XCR(I);
TW=T(I-1);
fW=b(I-1);
Xww=X(I-1);
TERRDE(I) = 1/(2*(XE-XP))*((TEE-TP)/(XEE-XP)-(TE-TW)/(XE-XW))...
* (((XP-Xe)^2-(XE-Xe)^2)/(XE-XP))...
+ (1/(Xee-Xe)*((TEE-TE)/(XEE-XE)-(TE-TP)/(XE-XP)) - 1/(Xe-Xw)*((TE-TP)/(XE-XP)-(TP-TW)/(XP-XW)))...
* 1/(6*(XE-XP))*(((XP-Xe)^3-(XE-Xe)^3)/(XE-XP));
TERRDW(I) = 1/(2*(XP-XW))*((TE-TW)/(XE-XW)-(TP-RBW)/(XP-Xww))...
* (((XW-Xw)^2-(XP-Xw)^2)/(XP-XW))...
+ (1/(Xe-Xw)*((TE-TP)/(XE-XP)-(TP-TW)/(XP-XW)) - 1/(Xw-Xww)*((TP-TW)/(XP-XW)-(TW-RBW)/(XW-Xww)))...
* 1/(6*(XP-XW))*(((XW-Xw)^3-(XP-Xw)^3)/(XP-XW));
% east -1
elseif I==N-1
XE=XCR(I+2);
TE=T(I+1);
fE=b(I+1);
Xee=X(I+2);
XW=XCR(I);
TW=T(I-1);
fW=b(I-1);
Xww=X(I-1);
XWW=XCR(I-1);
TWW=T(I-2);
TERRDE(I) = 1/(2*(XE-XP))*((RBE-TP)/(Xee-XP)-(TE-TW)/(XE-XW))...
* (((XP-Xe)^2-(XE-Xe)^2)/(XE-XP))...
+ (1/(Xee-Xe)*((RBE-TE)/(Xee-XE)-(TE-TP)/(XE-XP)) - 1/(Xe-Xw)*((TE-TP)/(XE-XP)-(TP-TW)/(XP-XW)))...
* 1/(6*(XE-XP))*(((XP-Xe)^3-(XE-Xe)^3)/(XE-XP));
TERRDW(I) = 1/(2*(XP-XW))*((TE-TW)/(XE-XW)-(TP-TWW)/(XP-XWW))...
* (((XW-Xw)^2-(XP-Xw)^2)/(XP-XW))...
+ (1/(Xe-Xw)*((TE-TP)/(XE-XP)-(TP-TW)/(XP-XW)) - 1/(Xw-Xww)*((TP-TW)/(XP-XW)-(TW-TWW)/(XW-XWW)))...
* 1/(6*(XP-XW))*(((XW-Xw)^3-(XP-Xw)^3)/(XP-XW));
% east
elseif I==N
XW=XCR(I);
TW=T(I-1);
fW=b(I-1);
Xww=X(I-1);
XWW=XCR(I-1);
TWW=T(I-2);
Dw = (XP-Xw)/(XP-XW);
Tw = Dw * TW + (1-Dw)*TP;
TERRDE(I) = 1/2*((RBE-TP)/(Xe-XP) - (RBE-Tw)/(Xe-Xw));
TERRDW(I) = 1/(2*(XP-XW))*((RBE-TW)/(Xe-XW)-(TP-TWW)/(XP-XWW))...
* (((XW-Xw)^2-(XP-Xw)^2)/(XP-XW))...
+ (1/(Xe-Xw)*((RBE-TP)/(Xe-XP)-(TP-TW)/(XP-XW)) - 1/(Xw-Xww)*((TP-TW)/(XP-XW)-(TW-TWW)/(XW-XWW)))...
* 1/(6*(XP-XW))*(((XW-Xw)^3-(XP-Xw)^3)/(XP-XW));
else
XW=XCR(I);
TW=T(I-1);
fW=b(I-1);
Xww=X(I-1);
XE=XCR(I+2);
TE=T(I+1);
fE=b(I+1);
Xee=X(I+2);
XWW=XCR(I-1);
TWW=T(I-2);
XEE=XCR(I+3);
TEE=T(I+2);
TERRDE(I) = 1/(2*(XE-XP))*((TEE-TP)/(XEE-XP)-(TE-TW)/(XE-XW))...
* (((XP-Xe)^2-(XE-Xe)^2)/(XE-XP))...
+ (1/(Xee-Xe)*((TEE-TE)/(XEE-XE)-(TE-TP)/(XE-XP)) - 1/(Xe-Xw)*((TE-TP)/(XE-XP)-(TP-TW)/(XP-XW)))...
* 1/(6*(XE-XP))*(((XP-Xe)^3-(XE-Xe)^3)/(XE-XP));
TERRDW(I) = 1/(2*(XP-XW))*((TE-TW)/(XE-XW)-(TP-TWW)/(XP-XWW))...
* (((XW-Xw)^2-(XP-Xw)^2)/(XP-XW))...
+ (1/(Xe-Xw)*((TE-TP)/(XE-XP)-(TP-TW)/(XP-XW)) - 1/(Xw-Xww)*((TP-TW)/(XP-XW)-(TW-TWW)/(XW-XWW)))...
* 1/(6*(XP-XW))*(((XW-Xw)^3-(XP-Xw)^3)/(XP-XW));
end
end
for I=1:N
TERR(I) = TERRS(I)-TERRDE(I)+TERRDW(I);
end
hold on;
plot(XC, TERR, 'rx-');
figure(5)
plot(XC, RES-TERR, 'x-');
title('RES - TE')