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comb1dflux.m
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comb1dflux.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;
BETA=0.5;
XMIN=0.0;
XMAX=1.0;
N=40; % KV's
X = linspace(XMIN, XMAX, N+1);
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*DX);
AW(I) = DIF/(DXW*DX);
AP(I) = -AE(I)-AW(I);
% UDS
DXeP = X(I+1)-XCR(I+1);
DXPw = XCR(I+1)-X(I);
AEKU(I) = 0;
AWKU(I) = -KONV/DX;
APKU(I) = KONV/DX;
% CDS
DCDSE = (X(I+1)-XCR(I+1))/DXE;
DCDSW = (XCR(I+1)-X(I))/DXW;
AEKC(I) = KONV/DX * DCDSE;
AWKC(I) = -KONV/DX * DCDSW;
APKC(I) = KONV/DX * ((1-DCDSE)-(1-DCDSW));
AKC(I) = KONV/DX;
end
% Gesamtgleichungssystem aufstellen
A = zeros(N);
b = zeros(N,1);
for I=1:N
b(I) = MSOL(XC(I));
% Hauptdiagonale
A(I,I) = AP(I) + (1-BETA)*APKU(I) + BETA*APKC(I);
% Westliche Nebendiagonale
if mod(I,N)==1
b(I) = b(I) - AW(I)*RBW + AKC(I)*RBW;
else
A(I, I-1) = AW(I) + (1-BETA)*AWKU(I) + BETA*AWKC(I);
end
% Östliche Nebendiagonale
if mod(I,N)==0
b(I) = b(I) - AE(I)*RBE - AKC(I)*RBE;
A(I, I) = AP(I) + BETA*APKC(I);
else
A(I, I+1) = AE(I) + (1-BETA)*AEKU(I) + BETA*AEKC(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
% für sin x +1, BETA=0.5
ERR5=4.7252901913e-02;
ERR10=2.0353266115e-02;
ERR20=9.4119189630e-03;
ERR40=4.5231320218e-03;
ERR80=2.2170064635e-03;
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
TE=zeros(1, N);
for I=1:N
b(I) = MSOL(XC(I));
end
for I=3:N-2
DX = X(I+1)-X(I);
TE(I) = (DX/24 * (b(I+1) - 2*b(I) + b(I-1)))... % TE_source
+ ((T(I+2)-3*T(I+1)+3*T(I)-T(I-1))/(24*DX))... % TE_e
- ((T(I+1)-3*T(I)+3*T(I-1)-T(I-2))/(24*DX))... % TE_w
- (1-BETA)*(3/8*T(I+1) - T(I)/4 - T(I-1)/8)... % TE_uds_e
+ (1-BETA)*(3/8*T(I) - T(I-1)/4 - T(I-2)/8)... % TE_uds_w
+ BETA*(T(I+1)-2*T(I)+T(I-1))/8 ... % TE_e_cds
+ BETA*(T(I+2)-2*T(I+1)+2*T(I-1)-T(I-2))/32 ... % TE_e_cds
- BETA*(T(I+1)-2*T(I)+T(I-1))/8 ...
+ BETA*(T(I+2)-2*T(I+1)+2*T(I-1)-T(I-2))/32;
TE(I) = TE(I)/DX;
end;
% TE Sonderfälle für Randvolumen
I=2;
DX = X(I+1)-X(I);
TE(I) = (DX/24 * (b(I+1) - 2*b(I) + b(I-1)))... % TE_source
+ ((T(I+2)-3*T(I+1)+3*T(I)-T(I-1))/(24*DX))... % TE_e
- ((T(I+1)-3*T(I)+4*T(I-1)-2*RBW)/(24*DX))... % TE_w
- (1-BETA)*(3/8*T(I+1) - T(I)/4 - T(I-1)/8)... % TE_uds_e
+ (1-BETA)*(3/8*T(I) + T(I-1)/8 - RBW/2); % TE_uds_w_rand
+ BETA*(T(I+1)-2*T(I)+T(I-1))/8 ... % TE_e_cds
+ BETA*(T(I+2)-2*T(I+1)+2*T(I-1)-2*RBW)/32 ... % TE_e_cds
- BETA*(T(I+1)-2*T(I)+T(I-1))/8 ...
+ BETA*(T(I+2)-2*T(I+1)+3*T(I-1)-2*RBW)/32;
TE(I) = TE(I)/DX;
I = N-1;
DX = X(I+1)-X(I);
TE(I) = (DX/24 * (b(I+1) - 2*b(I) + b(I-1)))... % TE_source
+ ((2*RBE-4*T(I+1)+3*T(I)-T(I-1))/(24*DX))... % TE_e
- ((T(I+1)-3*T(I)+3*T(I-1)-T(I-2))/(24*DX))... % TE_w
- (1-BETA)*(3/8*T(I+1) - T(I)/4 - T(I-1)/8)... % TE_uds_e
+ (1-BETA)*(3/8*T(I) - T(I-1)/4 - T(I-2)/8); % TE_uds_w
+ BETA*(T(I+1)-2*T(I)+T(I-1))/8 ... % TE_e_cds
+ BETA*(2*RBE-3*T(I+1)+2*T(I-1)-T(I-2))/32 ... % TE_e_cds
- BETA*(T(I+1)-2*T(I)+T(I-1))/8 ...
+ BETA*(2*RBE-3*T(I+1)+2*T(I-1)-T(I-2))/32;
TE(I) = TE(I)/DX;
hold on;
plot(XC, TE, 'rx-');
RESTE = RES-TE';
figure(5)
plot(XC(3:N-2), RESTE(3:N-2), 'x-')
xlabel('XC')
ylabel('RES-TE')