How to smooth or clear the peak of the curve at certain time level?

This is the tridaigonal system. here, i incoparate the time steps and it will stop at tolerance limit. and then i got the datas in T to plot graph in multiple time line. From begining all the curves coming downward(it has no problem). after the certain time level at 61th coloumn (in T workspace) it is coming peak. I have confused why its coming and how to clear or remove to get the smooth curve?

clc; close all; clear all;
%======================SPACEGRID====================================%
ymax=14; m=56; dy=ymax/m; y=dy:dy:ymax;      %'i'th row
xmax=1; n=20;  dx=xmax/n; x=dx:dx:xmax;     %'j'th column   
%=====================TIMEGRID======================================%
tmax=100; nt=500; dt=tmax/nt; t=dt:dt:tmax;      % time at 'j'
%=====================STEADYSTATEINPUTVALUES========================%
tol=1e-2; 
max_difference(1)=1;  max_Iteration=1;
umax_difference(1)=1; %umax_Iteration=1;
%======================INPUTVALUES===================================%
pr=6.2;del=0;   
%======================NFINPUTVALUES===============================%
PHI1=0; PHI2=0.; S=0;
rhoF=997.1; rhobetaF=20939.1; rhocpF=4166881; kF=0.613;  sF=0.05;
rhoC=8933; rhobetaC=14918.11; rhocpC=3439205; kC=401;  sC=59.6*10^6;           
rhoZ=5600; rhobetaZ=2413.6; rhocpZ=2773120; kZ=13;  sZ=1*10^-2;               
KNF=kF*kC+S-1*kF-S-1*PHI1*kF-kC/kC+S-1*kF+PHI1*kF-kC;
KHNF=KNF*kZ+S-1*KNF-S-1*PHI2*KNF-kZ/kZ+S-1*KNF+PHI2*KNF-kZ;
SNF=sF*sC+S-1*sF-S-1*PHI1*sF-sC/sC+S-1*sF+PHI1*sF-sC;
SHNF=SNF*sZ+S-1*SNF-S-1*PHI2*SNF-sZ/sZ+S-1*SNF+PHI2*SNF-sZ;
RHOHNF=1-PHI1*1-PHI2+PHI1*rhoC/rhoF+PHI2*rhoZ/rhoF;
RHOCPHNF=1-PHI1*1-PHI2+PHI1*rhocpC/rhocpF+PHI2*rhocpZ/rhocpF;
RHOBETAHNF=1-PHI1*1-PHI2+PHI1*rhobetaC/rhobetaF+PHI2*rhobetaZ/rhobetaF;
E1=1/1-PHI1^2.5*1-PHI2^2.5*1/RHOHNF;
E2=RHOBETAHNF/RHOHNF;
E3=SHNF/RHOHNF;
E4=1/pr*KHNF/KNF*1/RHOCPHNF;
E5=1/RHOCPHNF;
%====================INTIALIZATION=================================%
UOLD=zeros(m,nt);
VOLD=zeros(m,nt);
TNEW=0; TOLD=TNEW*ones(m,nt); TWALL=ones(1,length(t)); 
A=zeros([1,m]); 
B=A;
C=A;
D=A;
T=TOLD; U=UOLD; V=VOLD;
tic
%===================ENERGYEQUATION==============================%
while max_Iteration>tol
    for j=1:nt
        for i=1:m
            if j>i
                C(i)=dt*VOLD(i,j)/4*dy-dt*E4/(2*dy^2);
            elseif i>j
                A(i)=-dt*VOLD(i,j)/4*dy-dt*E4/(2*dy^2);
            elseif i==j
                B(i)=1+dt*UOLD(i,j)/2*dx+dt*E4/dy^2+dt*1/2*E5*del;
            end
        end
    end
    for j=2:nt
        for i=1:m-1
            if i==1
                D(i)=TOLD(i,j)-dt*UOLD(i,j)*(-T(i,j-1)+TOLD(i,j)-TOLD(i,j-1))/2*dx-dt*VOLD(i,j)*(TWALL(j)-TOLD(i-0,j))/4*dy+dt*E4*(TOLD(i-0,j)-2*TOLD(i,j)+TWALL(j)/2*dy^2)+(TOLD(i,j)/2*dt*E5*del)-dt*E4+E5*TWALL(j)/2*dy^2;
            elseif i==m-1
                D(i)=TOLD(i,j)-dt*UOLD(i,j)*(-T(i,j-1)+TOLD(i,j)-TOLD(i,j-1))/2*dx-dt*VOLD(i,j)*(TOLD(i+1,j)-TNEW)/4*dy+dt*E4*(TNEW-2*TOLD(i,j)+TOLD(i+1,j)/2*dy^2)+(TOLD(i,j)/2*dt*E5*del)-dt*(E4+E5)*TNEW/2*dy^2;
            else
                D(i)=TOLD(i,j)-dt*UOLD(i,j)*(-T(i,j-1)+TOLD(i,j)-TOLD(i,j-1))/2*dx-dt*VOLD(i,j)*(TOLD(i+1,j)-TOLD(i-1,j))/4*dy+dt*E4*(TOLD(i-1,j)-2*TOLD(i,j)+TOLD(i+1,j)/2*dy^2)+(TOLD(i,j)/2*dt*E5*del)-dt*TOLD(i,j)/2*dy^2;
            end
        end
        T(:,j)=TriDiag(A,B,C,D);
        dt=0.2+dt;
        TOLD=T;
        T(1,:)=TWALL(j);
        %====================STEADY STATE=================================%     
        max_difference(j)=max(abs(T(:,j)-T(:,j-1))./max(1,abs(T(:,j-1))));
        if max_difference(j)<tol
            break
        end
        max_Iteration=max_difference;
    end
end

function x = TriDiag(e,f,g,r)
%input
% e = subdiagonal vector
% f = diagonal vector
% g = superdiagonal vector
% r = right hand side vector
% output:
% x = solution vector


n=length(f);
for k = 2:n
    factor = e(k)/f(k-1);
    f(k) = f(k) - factor*g(k-1);
    r(k) = r(k) - factor*r(k-1);
end
x(n) = r(n)/f(n);
for k = n-1:-1:1
    x(k) = (r(k)-g(k)*x(k+1))/f(k);
end
for i = 1:length(x)
    fprintf('\n%d = %f\n', i, x(i));
end

Kindly, help me in this hardest level.

Thank you.

Answer 1

[Karen]

Hey! It sounds like you’re seeing an unusual peak in your tridiagonal system’s graph around the 61st column. This could be due to boundary conditions or even some numerical instability as the time steps progress.