Let's assume that the temperature distribution inside a pipe is governed by the following one dimensional unsteady hyperbolic partial-differential-equation

**T**_{t}+ u T_{x}= 0

Consider negligible heat diffusion and the initial velocity u is, 0.1 cm/s. The boundary conditions are described below:

**T(x, 0) = 200 x; 0 ≤ x ≤ 0.5**

**T(x, 0) = 200 (1 – x); 0.5 ≤ x ≤ 1**

Develop an algorithm in MATLAB to solve this problem using the second-order approximation in time and the second-order upwind approximation in space. Consider,

*Δx = 0.05 cm; Δt = 0.05.*

The following MATLAB program develops the second order upwind method to solve the given one-dimensional unsteady hyperbolic convection equation.

**% Hyperbolic equation - Convection equation: Upwind Method**

**close**

**all**

**;**

**clc;**

**% Input Properties from Problem**

**L=1.0;**

**% (dimension in cm)**

**Lmax=5.0;**

**alpha=0.01;**

**u=0.1;**

**% (dimension in cm/s)**

**Tt=10;**

**% Simulation Time Period**

**% Mesh/Grid size**

**dx=0.05;**

**dt=0.05;**

**c=u*dt/dx;**

**% Depends on dt values**

**d=alpha*((dt/dx)^2.0);**

**% Matrix Parameters**

**imax=(Lmax/dx)+1;**

**% Array dimension along the width**

**tmax=(Tt/dt);**

**% Array dimension along the height**

**ndim=imax-2;**

**tdim=tmax-2;**

**% Total no of iterations**

**iteration=10000;**

**% Convergence Criterion**

**tolerance=0.000001;**

**% Array and Variables**

**% a = Matrix of Coefficients A(i,j)**

**% b = Right Side Vector, b(i)**

**% x = Solution Vector, x(i)**

**% Initial Condition**

**f(:,1)=0.0;**

**for**

**i=1:imax-2**

**x(i)=(i-1)*dx;**

**if**

**x(i)<=1.0**

**f(i,1)=0.0;**

**end**

**if**

**x(i)>1.0 && x(i)<=1.5**

**f(i,1)=200*(x(i)-1.0);**

**end**

**if**

**x(i)>1.5 && x(i)<=2.0**

**f(i,1)=200*(L-x(i)+1.0);**

**end**

**if**

**x(i)>2.0**

**f(i,1)=0;**

**end**

**end**

**fprintf(**

**'imax = %f\n'**

**,imax);**

**fprintf(**

**'c=%f\n'**

**,c);**

**% Upwind Method**

**for**

**t=1:tmax**

**time(t+1)=(t)*dt;**

**for**

**i=2:imax-2**

**% Second Order Approximation**

**f(i,t+1)=f(i,t)-c*(f(i,t)-f(i-1,t))-0.5*c*(1.0-c)*(f(i,t)-2.0*f(i-1,t)+f(i-2,t));**

**end**

**end**

**hold**

**on**

**plot(x,f(:,1))**

**plot(x,f(:,find(time==1)),**

**'o-'**

**)**

**plot(x,f(:,find(time==5)),**

**'o-'**

**)**

**plot(x,f(:,find(time==10)),**

**'o-'**

**)**

**xlabel(**

**'x, cm'**

**)**

**ylabel(**

**'Temperature, C'**

**)**

**legend(**

**'t=0s'**

**,**

**'t=1s'**

**,**

**'t=5s'**

**,**

**'t=10s'**

**)**

**title (**

**'c=0.1'**

**)**

__Program Output:__

__Program Output:__

__Second-order approximation in time and second-order upwind approximation in space:__

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