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comb2d_cds_orth.m
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comb2d_cds_orth.m
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clc
clear all
close all
SOL=@(x,y) sin(pi*x)*sin(pi*y);
MSOL=@(x,y) -2*pi^2*sin(pi*x)*sin(pi*y) + pi*cos(pi*x)*sin(pi*y) + pi*sin(pi*x)*cos(pi*y);
%SOL=@(x,y) sin(pi/2*x)*sin(pi/2*y);
%MSOL=@(x,y) pi/2*cos(pi/2*x)*cos(pi/2*y) -pi/2*sin(pi/2*x)*sin(pi/2*y) -pi^2/2*sin(pi/2*x)*cos(pi/2*y);
DIF=1.0;
KONV=1.0;
XMIN=0.0;
XMAX=1.0;
YMIN=0.0;
YMAX=1.0;
ALPHAX=0.9;
ALPHAY=0.9;
N=40; % KV's in einer Koordinatenrichtung, macht N^2 KV gesamt
NN=N*N;
X = zeros(1,N+1);
for I=1:N+1
X(I) = XMIN + (ALPHAX^(I-1)-1)/(ALPHAX^N-1)*(XMAX-XMIN);
end
Y = zeros(1,N+1);
for I=1:N+1
Y(I) = YMIN + (ALPHAY^(I-1)-1)/(ALPHAY^N-1)*(YMAX-YMIN);
end
NDIVS = 3;
% N halbieren
%for j=1:NDIVS
%idx =1;
%X2=0;
%for i=1:length(X)+1
%if mod(i, 2)==1
%X2(idx)=X(i);
%idx=idx+1;
%end
%end
%X=X2;
%N=length(X)-1;
%end
%for j=1:NDIVS
%idx =1;
%Y2=0;
%for i=1:length(Y)+1
%if mod(i, 2)==1
%Y2(idx)=Y(i);
%idx=idx+1;
%end
%end
%Y=Y2;
%N=length(Y)-1;
%end
%NN=N*N;
XC = (X(1:N)+X(2:N+1))/2;
YC = (Y(1:N)+Y(2:N+1))/2;
XCR = [XMIN, XC, XMAX];
YCR = [YMIN, YC, YMAX];
%%% ANALYTISCHE LÖSUNG
TA = zeros(N, N);
for I=1:N
for J=1:N
TA(I, J)=SOL(XC(I), YC(J));
end
end
figure(1)
surf(XC, YC, TA');
title('Analytische Loesung')
xlabel('X')
ylabel('Y')
%%% FVM Lösung
% Randbedingungen
for I=1:N RBS(I)=SOL(XC(I), 0); end
for I=1:N RBN(I)=SOL(XC(I), 1); end
for I=1:N RBE(I)=SOL(1, YC(I)); end
for I=1:N RBW(I)=SOL(0, YC(I)); end
% Koeffizienten speichern
AP = zeros(N);
AE = zeros(N);
AN = zeros(N);
AW = zeros(N);
AS = zeros(N);
for I=1:N
for J=1:N
DX = X(I+1)-X(I);
DY = Y(J+1)-Y(J);
DXE = XCR(I+2)-XCR(I+1);
DXW = XCR(I+1)-XCR(I);
DYN = YCR(J+2)-YCR(J+1);
DYS = YCR(J+1)-YCR(J);
AE(I,J) = DIF*DY/DXE;
AW(I,J) = DIF*DY/DXW;
AN(I,J) = DIF*DX/DYN;
AS(I,J) = DIF*DX/DYS;
AP(I,J) = -AE(I,J)-AN(I,J)-AW(I,J)-AS(I,J);
DCDSE = (X(I+1)-XCR(I+1))/DXE;
DCDSW = (XCR(I+1)-X(I))/DXW;
DCDSN = (Y(J+1)-YCR(J+1))/DYN;
DCDSS = (YCR(J+1)-Y(J))/DYS;
AEK(I,J) = KONV * DCDSE * DY;
AWK(I,J) =-KONV * DCDSW * DY;
ANK(I,J) = KONV * DCDSN * DX;
ASK(I,J) =-KONV * DCDSS * DX;
APK(I,J) = KONV*((1-DCDSE)-(1-DCDSW))*DY+KONV*((1-DCDSN)-(1-DCDSS))*DX;
AK(I,J) = KONV;
end
end
% Gesamtgleichungssystem aufstellen
A = zeros(NN);
b = zeros(NN, 1);
for J=1:N
for I=1:N
IDX = (J-1)*N + I;
DX = X(I+1)-X(I);
DY = Y(J+1)-Y(J);
b(IDX) = MSOL(XC(I),YC(J))*DX*DY;
% Hauptdiagonale
A(IDX, IDX) = AP(I,J) + APK(I,J);
A2(IDX,IDX) = APK(I,J);
% Westliche Nebendiagonale
if mod(IDX,N)==1
b(IDX) = b(IDX) - AW(I,J)*RBW(J) + AK(I,J)*RBW(J);
else
A(IDX, IDX-1) = AW(I,J) + AWK(I,J);
end
% Östliche Nebendiagonale
if mod(IDX,N)==0
b(IDX) = b(IDX) - AE(I,J)*RBE(J) - AK(I,J)*RBE(J);
else
A(IDX, IDX+1) = AE(I,J) + AEK(I,J);
end
% Nördliche Nebendiagonale
if IDX > NN-N
b(IDX) = b(IDX) - AN(I,J)*RBN(I) + AK(I,J)*RBN(I);
else
A(IDX, IDX+N) = AN(I,J) + ANK(I,J);
end
% Südliche Nebendiagonale
if IDX <= N
b(IDX) = b(IDX) - AS(I,J)*RBS(I) - AK(I,J)*RBS(I);
else
A(IDX, IDX-N) = AS(I,J) + ASK(I,J);
end
end
end
t=A\b;
T=reshape(t,N,N);
% mit Randwerten
T2 = zeros(N+2);
T2(2:N+1,2:N+1) = T;
figure(2)
surf(XC, YC, T');
title('Numerische Loesung')
xlabel('X')
ylabel('Y')
%%% Lösungsfehler berechnen
SERR=0.0;
ERR=zeros(N);
for I=1:N
for J=1:N
ERR(I,J)=T(I, J)-TA(I, J);
SERR=SERR+ERR(I,J)^2;
end
end
SERR=sqrt(SERR/NN);
figure(3)
surf(XC, YC, ERR');
title('Loesungsfehler')
fprintf('Summierter Fehler %16.10e NN=%g\n', SERR, NN);
%%% ORDNUNG BESTIMMEN
%T3
ERR5=5.7803454106e-02;
ERR10=9.2055921584e-03;
ERR20=2.0604562105e-03;
ERR40=5.0728031553e-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
s=zeros(N);
for I=1:N
for J=1:N
s(I, J)=SOL(XC(I), YC(J));
end
end
s=reshape(s, NN, 1);
RES=A*s-b;
RES2 = reshape(RES, N, N);
figure(4)
surf(XC, YC, RES2');
xlabel('XC')
ylabel('YC')
zlabel('RES')
title('Residuum')
%%% Truncation Error berechnen
% b ohne Randwerte
TERR=zeros(N);
b=zeros(N);
for I=1:N
for J=1:N
DX = X(I+1)-X(I);
DY = Y(J+1)-Y(J);
b(I,J) = MSOL(XC(I),YC(J))*DX*DY;
end
end
for I=3:N-2
for J=3:N-2
% benötigte Werte zwischenspeichern
XEE=XCR(I+3);
XE=XCR(I+2);
XP=XCR(I+1);
XW=XCR(I);
XWW=XCR(I-1);
Xe=X(I+1);
Xee=X(I+2);
Xw=X(I);
Xww=X(I-1);
DX = Xe-Xw;
YNN=YCR(J+3);
YN=YCR(J+2);
YP=YCR(J+1);
YS=YCR(J);
YSS=YCR(J-1);
Yn=Y(J+1);
Ynn=Y(J+2);
Ys=Y(J);
Yss=Y(J-1);
DY = Yn-Ys;
fE=b(I+1,J);
fP=b(I,J);
fW=b(I-1,J);
fN=b(I,J+1);
fS=b(I,J-1);
TEE=T(I+2,J);
TE=T(I+1,J);
TP=T(I,J);
TW=T(I-1,J);
TWW=T(I-2,J);
TN=T(I,J+1);
TNN=T(I,J+2);
TS=T(I,J-1);
TSS=T(I,J-2);
DX = X(I+1)-X(I);
DY = Y(J+1)-Y(J);
% Source Term
TERRSO = 1/2*((fE-fP)/((XE-XP)*DX) - (fP-fW)/((XP-XW)*DX))...
* ((Xe-XP)^3 - (Xw-XP)^3)/3*DY...
+ 1/2*((fN-fP)/((YN-YP)*DY) - (fP-fS)/((YP-YS)*DY))...
* ((Yn-YP)^3 - (Ys-YP)^3)/3*DX;
TERRE = 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));
TERRW = 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));
TERRN = 1/(2*(YN-YP))*((TNN-TP)/(YNN-YP)-(TN-TS)/(YN-YS))...
* (((YP-Yn)^2-(YN-Yn)^2)/(YN-YP))...
+ (1/(Ynn-Yn)*((TNN-TN)/(YNN-YN)-(TN-TP)/(YN-YP)) - 1/(Yn-Ys)*((TN-TP)/(YN-YP)-(TP-TS)/(YP-YS)))...
* 1/(6*(YN-YP))*(((YP-Yn)^3-(YN-Yn)^3)/(YN-YP));
TERRS = 1/(2*(YP-YS))*((TN-TS)/(YN-YS)-(TP-TSS)/(YP-YSS))...
* (((YS-Ys)^2-(YP-Ys)^2)/(YP-YS))...
+ (1/(Yn-Ys)*((TN-TP)/(YN-YP)-(TP-TS)/(YP-YS)) - 1/(Ys-Yss)*((TP-TS)/(YP-YS)-(TS-TSS)/(YS-YSS)))...
* 1/(6*(YP-YS))*(((YS-Ys)^3-(YP-Ys)^3)/(YP-YS));
TERRKE = 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 = 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)));
TERRKN = 1/(2*(Yn-Ys))*((TN-TP)/(YN-YP) - (TP-TS)/(YP-YS))*(Yn-YN)*(Yn-YP)...
+ 1/6*(Yn-YP)/DY*((Yn-YP)^2-(YN-YP)^2)...
* (1/(YN-YP)*((TNN-TP)/(YNN-YP)-(TN-TS)/(YN-YS)) - 1/(YP-YS)*((TN-TS)/(YN-YS) - (TP-TSS)/(YP-YSS)));
TERRKS = 1/(2*(Yn-Ys))*((TN-TP)/(YN-YP) - (TP-TS)/(YP-YS))*(Ys-YS)*(Ys-YP)...
+ 1/6*(Ys-YP)/DY*((Ys-YP)^2-(YS-YP)^2)...
* (1/(YN-YP)*((TNN-TP)/(YNN-YP)-(TN-TS)/(YN-YS)) - 1/(YP-YS)*((TN-TS)/(YN-YS) - (TP-TSS)/(YP-YSS)));
TERR(I,J) = TERRSO - TERRE + TERRW - TERRN + TERRS...
-TERRKE +TERRKW -TERRKN +TERRKS;
end
end
figure(5)
surf(XC, YC, TERR');
title('TE');
RESTE = RES2-TERR;
figure(6)
surf(XC(3:N-2), YC(3:N-2), RESTE(3:N-2, 3:N-2)');
title('RES-TE');