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comb1dcds_orth.m
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comb1dcds_orth.m
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clc
clear all
close all
SOL=@(x) sin(pi*x)+1;
MSOL=@(x) pi*cos(pi*x)-pi^2*sin(pi*x);
%SOL=@(x) cos(pi*x)-1;
%MSOL=@(x) -pi*sin(pi*x)-pi^2*cos(pi*x);
DIF=1.0;
KONV=1.0;
ALPHA=0.9;
XMIN=0.0;
XMAX=1.0;
N=80; % KV's
X = zeros(1,N+1);
for I=1:N+1
X(I) = XMIN + (ALPHA^(I-1)-1)/(ALPHA^N-1)*(XMAX-XMIN);
end
% N halbieren
%for j=1:1
%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
DX = X(I+1)-X(I);
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);
DCDSE = (X(I+1)-XCR(I+1))/DXE;
DCDSW = (XCR(I+1)-X(I))/DXW;
AEK(I) = KONV * DCDSE;
AWK(I) = -KONV * DCDSW;
APK(I) = KONV * ((1-DCDSE)-(1-DCDSW));
AK(I) = KONV;
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) + APK(I);
% Westliche Nebendiagonale
if mod(I,N)==1
b(I) = b(I) - AW(I)*RBW + AK(I)*RBW;
else
A(I, I-1) = AW(I) + AWK(I);
end
% Östliche Nebendiagonale
if mod(I,N)==0
b(I) = b(I) - AE(I)*RBE - AK(I)*RBE;
else
A(I, I+1) = AE(I) + AEK(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, N);
%%% ORDNUNG BESTIMMEN
ERR5=1.2662511400e-01;
ERR10=1.9068695077e-02;
ERR20=4.0741696509e-03;
ERR40=9.9564249986e-04;
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 );
%%% 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);
TERRKE=zeros(N,1);
TERRKW=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));
TERRKE(I) = 1/(2*(Xe-Xw))*((TE-TP)/(XE-XP) - (TP-RBW)/(XP-Xw))*(Xe-XE)*(Xe-XP)...
+ 1/6*(Xe-XP)/DX*((Xe-XP)^2-(XE-XP)^2)...
* (1/(XE-XP)*((TEE-TP)/(XEE-XP)-(TE-RBW)/(XE-Xw)) - 1/(XP-Xw)*((TE-RBW)/(XE-Xw) - (TP-RBW)/(XP-Xw)));
TERRKW(I) = 0;
% 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));
TERRKE(I) = 1/(2*(Xe-Xw))*((TE-TP)/(XE-XP) - (TP-TW)/(XP-XW))*(Xe-XE)*(Xe-XP)...
+ 1/6*(Xe-XP)/DX*((Xe-XP)^2-(XE-XP)^2)...
* (1/(XE-XP)*((TEE-TP)/(XEE-XP)-(TE-TW)/(XE-XW)) - 1/(XP-XW)*((TE-TW)/(XE-XW) - (TP-RBW)/(XP-Xww)));
TERRKW(I) = 1/(2*(Xe-Xw))*((TE-TP)/(XE-XP) - (TP-TW)/(XP-XW))*(Xw-XW)*(Xw-XP)...
+ 1/6*(Xw-XP)/DX*((Xw-XP)^2-(XW-XP)^2)...
* (1/(XE-XP)*((TEE-TP)/(XEE-XP)-(TE-TW)/(XE-XW)) - 1/(XP-XW)*((TE-TW)/(XE-XW) - (TP-RBW)/(XP-Xww)));
% 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))*((TEE-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));
TERRKE(I) = 1/(2*(Xe-Xw))*((TE-TP)/(XE-XP) - (TP-TW)/(XP-XW))*(Xe-XE)*(Xe-XP)...
+ 1/6*(Xe-XP)/DX*((Xe-XP)^2-(XE-XP)^2)...
* (1/(XE-XP)*((RBE-TP)/(Xee-XP)-(TE-TW)/(XE-XW)) - 1/(XP-XW)*((TE-TW)/(XE-XW) - (TP-TWW)/(XP-XWW)));
TERRKW(I) = 1/(2*(Xe-Xw))*((TE-TP)/(XE-XP) - (TP-TW)/(XP-XW))*(Xw-XW)*(Xw-XP)...
+ 1/6*(Xw-XP)/DX*((Xw-XP)^2-(XW-XP)^2)...
* (1/(XE-XP)*((RBE-TP)/(Xee-XP)-(TE-TW)/(XE-XW)) - 1/(XP-XW)*((TE-TW)/(XE-XW) - (TP-TWW)/(XP-XWW)));
% 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));
TERRKE(I) = 0;
TERRKW(I) = 1/(2*(Xe-Xw))*((RBE-TP)/(Xe-XP) - (TP-TW)/(XP-XW))*(Xw-XW)*(Xw-XP)...
+ 1/6*(Xw-XP)/DX*((Xw-XP)^2-(XW-XP)^2)...
* (1/(Xe-XP)*((RBE-TP)/(Xe-XP)-(RBE-TW)/(Xe-XW)) - 1/(XP-XW)*((RBE-TW)/(Xe-XW) - (TP-TWW)/(XP-XWW)));
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));
TERRKE(I) = 1/(2*(Xe-Xw))*((TE-TP)/(XE-XP) - (TP-TW)/(XP-XW))*(Xe-XE)*(Xe-XP)...
+ 1/6*(Xe-XP)/DX*((Xe-XP)^2-(XE-XP)^2)...
* (1/(XE-XP)*((TEE-TP)/(XEE-XP)-(TE-TW)/(XE-XW)) - 1/(XP-XW)*((TE-TW)/(XE-XW) - (TP-TWW)/(XP-XWW)));
TERRKW(I) = 1/(2*(Xe-Xw))*((TE-TP)/(XE-XP) - (TP-TW)/(XP-XW))*(Xw-XW)*(Xw-XP)...
+ 1/6*(Xw-XP)/DX*((Xw-XP)^2-(XW-XP)^2)...
* (1/(XE-XP)*((TEE-TP)/(XEE-XP)-(TE-TW)/(XE-XW)) - 1/(XP-XW)*((TE-TW)/(XE-XW) - (TP-TWW)/(XP-XWW)));
end
end
for I=1:N
TERR(I) = TERRS(I)-TERRDE(I)+TERRDW(I)-TERRKE(I)+TERRKW(I);
end
hold on;
plot(XC, TERR, 'rx-');
RESTE = RES-TERR;
figure(5)
plot(XC(1:N), RESTE(1:N), 'x-')
xlabel('XC')
ylabel('RES-TE')