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zE^!VaA$Ar~c5ZPFE^#hV0X|VSCLu^S9PC`8ES#cTEKD3i%=~l7|@rgGA>Om z1ry6dVJ{A2Qp2pBDrfUc`$E4(PRWLa+C(TbOy?<-fy1vmz?jSW0AE sPNpFy@&?v&$NYb0K#bt%WZ>ZB>R@7uz{<+S#>&otKu#{EAdc{V08^SS@c;k- literal 0 HcmV?d00001 From f7b816c68b90f170175308f00a64e455a7ef2f59 Mon Sep 17 00:00:00 2001 From: Moonwalker777 <73292958+Moonwalker777@users.noreply.github.com> Date: Fri, 13 May 2022 12:48:34 +0200 Subject: [PATCH 2/2] Riccardos Matlab Files --- Exercise_03/Dirichlet_EA.m | 59 ++++++++++++++++++++++++ Exercise_03/Mixed_EA.m | 63 ++++++++++++++++++++++++++ Exercise_03/Mixed_EI.m | 62 +++++++++++++++++++++++++ Exercise_03/NSSC_1.m | 93 ++++++++++++++++++++++++++++++++++++++ 4 files changed, 277 insertions(+) create mode 100755 Exercise_03/Dirichlet_EA.m create mode 100755 Exercise_03/Mixed_EA.m create mode 100755 Exercise_03/Mixed_EI.m create mode 100755 Exercise_03/NSSC_1.m diff --git a/Exercise_03/Dirichlet_EA.m b/Exercise_03/Dirichlet_EA.m new file mode 100755 index 0000000..1243748 --- /dev/null +++ b/Exercise_03/Dirichlet_EA.m @@ -0,0 +1,59 @@ +function [x,t,u] = Dirichlet_EA(L,N,T,K,c1,c2,f,u0,D) +% ---- Solution of heat equation ---- +% u_t - u_xx = f in the interval [-L,L] (doesn't matter if we change it) +% Dirichlet BC +% +% ----------------------------------------------- +% Sintassi: +% [x,t,u]=calore_template(L,N,T,K,c1,c2,fun,u0) +% +% Input: +% L Half of the width (-L,L) +% N number of intervals in (-L,L) +% T max time (0,T) +% K number of intervals in (0,T) +% c1 Dirichlet BC in x=-L +% c2 Dirichlet BC in x=L +% f force function +% u0 initial condition in t=0 +% +% Output: +% x vector of the spatial nodes +% t vector of the time nodes +% u numeric solution of the problem + +% discretisation step in time and space +h=L/N; %space +tau=T/K; %time +% Initialization of t +t=linspace(0,T,K+1)'; +% initialization of x +x=linspace(0,L,N+1); + +% Initialization of the matrix solution u +u=zeros(N+1,K+1); +% Intial conditions +u(:,1)=u0(x); + +% BC +u(1,:)=c1(t); +u(end,:)=c2(t); + +% Creation of the matrix A +% We obtained this matrix using the Central Discretization method +e=ones(N-1,1); +A=spdiags([-e,2*e,-e],[-1,0,1],N-1,N-1)/(h^2); +I=speye(N-1,N-1); + +% Compute the solution +% What we are doing here is to compute the solution in the interval for +% each time k +for k=1:K + % Assembly of the force + F=f(x(2:end-1),t(k)); + % Correction of the force using the BC + F(1)=F(1) + D*c1(t(k))/(h^2); + F(end)=F(end) + D*c2(t(k))/(h^2); + % Solution using Eulero forward + u(2:end-1,k+1) = ((I - tau*D*A)*u(2:end-1,k))' + tau*F; +end diff --git a/Exercise_03/Mixed_EA.m b/Exercise_03/Mixed_EA.m new file mode 100755 index 0000000..1ac4aee --- /dev/null +++ b/Exercise_03/Mixed_EA.m @@ -0,0 +1,63 @@ +function [x,t,u] = Mixed_EA(L,N,T,K,c1,c2,f,u0,D) +% ---- Solution of heat equation ---- +% u_t - u_xx = f in the interval [-L,L] (doesn't matter if we change it) +% Dirichlet BC +% +% ----------------------------------------------- +% Sintassi: +% [x,t,u]=calore_template(L,N,T,K,c1,c2,fun,u0) +% +% Input: +% L Half of the width (-L,L) +% N number of intervals in (-L,L) +% T max time (0,T) +% K number of intervals in (0,T) +% c1 Dirichlet BC in x=-L +% c2 Dirichlet BC in x=L +% f force function +% u0 initial condition in t=0 +% +% Output: +% x vector of the spatial nodes +% t vector of the time nodes +% u numeric solution of the problem + +% discretisation step in time and space +h=L/N; %space +tau=T/K; %time +% Initialization of t +t=linspace(0,T,K+1)'; +% initialization of x +x=linspace(0,L,N+1); + +% Initialization of the matrix solution u +u=zeros(N+1,K+1); +% Intial conditions +u(:,1)=u0(x); + +% BC Dirichlet +u(1,:)=c1(t); + +% Creation of the matrix A +% We obtained this matrix using the Central Discretization method +e=ones(N-1,1); +A=spdiags([-e,2*e,-e],[-1,0,1],N-1,N-1)/(h^2); +%modify the matrix such that we can use the noimann condition to calculate +%the last node. +%In this case we want to use a second order decentralized approximation +A(end,end-1)=-2/(h^2); +I=speye(N-1,N-1); + +% Compute the solution +% What we are doing here is to compute the solution in the interval for +% each time k +for k=1:K + % Assembly of the force + F=f(x(2:end-1),t(k)); + % Correction of the force using the BC + F(1)=F(1) + D*c1(t(k))/(h^2); + F(end)=F(end) + 2*h*c2; + % Solution using Eulero forward + u(2:end-1,k+1) = ((I - D*tau*A)*u(2:end-1,k))' + tau*F; + u(end,k+1)=u(end-2,k+1) + c2*2*h*D; +end diff --git a/Exercise_03/Mixed_EI.m b/Exercise_03/Mixed_EI.m new file mode 100755 index 0000000..635f458 --- /dev/null +++ b/Exercise_03/Mixed_EI.m @@ -0,0 +1,62 @@ +function [x,t,u] = Mixed_EI(L,N,T,K,c1,c2,f,u0,D) +% ---- Risoluzione dell'equazione del calore ---- +% u_t - u_xx = f nell'intervallo [-L,L] +% con condizioni al bordo di Dirichlet +% e condizioni iniziali. +% ----------------------------------------------- +% Sintassi: +% [x,t,u]=calore_template(L,N,T,K,c1,c2,fun,u0) +% +% Input: +% L semiampiezza intervallo spaziale (-L,L) +% N numero di sottointervalli in (-L,L) +% T estremo finale intervallo temporale (0,T) +% K numero di sottointervalli in (0,T) +% c1 funzione che descrive la condizione di Dirichlet in x=-L +% c2 funzione che descrive la condizione di Dirichlet in x=L +% f funzione che descrive il termine noto dell'equazione +% u0 funzione che descrive la condizione iniziale in t=0 +% +% Output: +% x vettore dei nodi spaziali +% t vettore dei nodi temporali +% u soluzione numerica +% del problema + +% Calcolo passo di discretizzazione in spazio e tempo +h=L/N; +tau=T/K; +% Inizializzazione del vettore t +t=linspace(0,T,K+1); +% Inizializzazione del vettore x +x=linspace(0,L,N+1); + +% Inizializzazione della matrice soluzione u +u=zeros(N+1,K+1); +% Condizione iniziale +u(:,1)=u0(x); + +% BC Dirichlet +u(1,:)=c1(t); + +% Creation of the matrix A +% We obtained this matrix using the Central Discretization method +e=ones(N-1,1); +A=spdiags([-e,2*e,-e],[-1,0,1],N-1,N-1)/(h^2); +%modify the matrix such that we can use the noimann condition to calculate +%the last node. +%In this case we want to use a second order decentralized approximation +A(end,end-1)=-2/(h^2); +I=speye(N-1,N-1); + + +for k=1:K + % Assemblaggio termine noto + F=f(x(2:end-1),t(k+1)); + % Correzione del termine noto con le condizioni al bordo + F(1)=F(1) + c1(t(k+1))/(h^2); + F(end)=c2*h*2; + % Risoluzione del problema + u(2:end-1,k+1) = ((I+tau*A)\u(2:end-1,k))' + tau*F; + u(end,k+1)=u(end-2,k+1) + c2*2*h; +end diff --git a/Exercise_03/NSSC_1.m b/Exercise_03/NSSC_1.m new file mode 100755 index 0000000..f548749 --- /dev/null +++ b/Exercise_03/NSSC_1.m @@ -0,0 +1,93 @@ +%% eulero forward DIrichlet +L=pi; +T=2; +f=@(x,t) 0*x.*t; +c1=@(t) 1+0*t; +c2=@(t) 0*t; +u0=@(x) 0*x; +D=0.5; +%uex=@(x,t) cos(x).*exp(t); + +N=10; +K=200; +[x,t,u]=Dirichlet_EA(L,N,T,K,c1,c2,f,u0,D); + +figure(1) +for ii=1:K+1 + plot(x,u(:,ii)'); + xlim([0 L]) + pause(0.05); +end + +space=linspace(0,L,101); +time=linspace(0,T,201); +[xx,yy]=meshgrid(time,space); +%exsol=uex(yy,xx); +figure(2) +mesh(t,x,u) +%figure(2) +%mesh(xx,yy,exsol) + +%% Eulero forward Mixed BC + +L=2*pi; +T=5; +f=@(x,t) 0*x.*t; +c1=@(t) 1+0*t; +c2=0; +u0=@(x) 0*x; +D=1.1; +%uex=@(x,t) cos(x).*exp(t); + +N=25; +K=200; +[x,t,u]=Mixed_EA(L,N,T,K,c1,c2,f,u0,D); + +figure(1) +for ii=1:K+1 + plot(x,u(:,ii)'); + xlim([0 L]) + ylim([0 1.5]) + pause(0.02); +end + +space=linspace(0,L,101); +time=linspace(0,T,201); +[xx,yy]=meshgrid(time,space); +%exsol=uex(yy,xx); +figure(2) +mesh(t,x,u) +%figure(2) +%mesh(xx,yy,exsol) + +%% Eulero Backward Mixed BC + +L=2*pi; +T=5; +f=@(x,t) 0*x.*t; +c1=@(t) 1+0*t; +c2=0; +u0=@(x) 0*x; +D=1; +%uex=@(x,t) cos(x).*exp(t); + +N=25; +K=200; +[x,t,u]=Mixed_EI(L,N,T,K,c1,c2,f,u0,D); + +figure(1) +for ii=1:K+1 + plot(x,u(:,ii)'); + xlim([0 L]) + ylim([0 1.5]) + pause(0.02); +end + +space=linspace(0,L,101); +time=linspace(0,T,201); +[xx,yy]=meshgrid(time,space); +%exsol=uex(yy,xx); +figure(2) +mesh(t,x,u) +%figure(2) +%mesh(xx,yy,exsol)

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