ME3275_project1.ipynb

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    "Project 1 ME3275\n",
    "\n",
    "Xinyu Zhao, Fall 2023\n",
    "\n",
    "A typical transport equation comprises a transient term, a convection (or advection to avoid confusion with heat transfer terminology) term, a diffusion term and a source term. We start from the simpliest problem in the first project: a 1-D advection of a wave form. The wave form can be acoustic waves or any other signals physically, and can take an initial shape of a step function or a sine function or anything else. In this project, we will transport a step function and the governing equation takes the following form: "
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    "$$\n",
    "\\frac{\\partial u} {\\partial t} + c \\frac{\\partial u}{\\partial x}  = 0~~(1)\n",
    "$$"
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    "The equation is a linear partial differential equation that has similar properties as a hyperbolic equation. $c$ is a constant here, $t$ indicates time, and $x$ is distance. The goal is to solve for $u(x,t)$, which requires one boundary condition and initial condition.  Let the initial condition be $u(x,0) = u_0(x)$. Then the exact solution of the equation is $u(x,t)=u_0(x-ct)$. "
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    "Consider a computational domain of $L$ = 2 m. $N$=41 grid points (including the two boundary points)\n",
    "will be used to discretize the 1-D computational domain, with equal spacing (i.e., $\\Delta x$ =\n",
    "$L/(N-1)$). We discretize this equation in both space and time. Consider discretizing the\n",
    "spatial coordinate $x$ into points that we index from i=0 to N, and stepping in discrete time\n",
    "interval of size $\\Delta t$ = 0.025 s, and $c$ is a constant of 1 m/s. Applying the first-order explicit scheme in time and first-order backward difference in space, the discretized equation at location $i$ and time $n$ becomes:\r",
    "$$\n",
    "(u^{n+1}_i - u^{n}_i)/\\Delta t + c(u^n_i - u^n_{i-1})/\\Delta x = 0~~(2)\n",
    "$$\n",
    "where $n$ and $n+1$ are two consecutive steps in time, while $i-1$ and $i$ are two neighboring points of the discretized $x$ coordinate. Given the initial conditions, the $\\underline{only}$ unknown in this equation is $u^{n+1}_i$. "
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   "source": [
    "Question (a) Re-arrange Equation (2), and obtain an explicit solution for $u^{n+1}_i$ as a function of all other terms in the equation."
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   "source": [
    "Question (b) Set the initial conditions of the velocity (i.e., $u_0$) using the following code. Add your own plotting code to visualize the initial shape of the velocity $u$."
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    "import numpy as np                      #here we load numpy as in HW1\n",
    "import matplotlib as plt              #here we load matplotlib as in HW1\n",
    "import time, sys                        #we load some additional functions"
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    "## Set up your problem: geometry, mesh, and constants\n",
    "nx = 41; # total number of grid points\n",
    "L = 2.0; # meter\n",
    "dx = L/(nx-1); # grid spacing\n",
    "nt = 25; # total number of time steps\n",
    "dt = .025; # second\n",
    "c = 1; # m/s\n",
    "\n",
    "## initialize the velocity field\n",
    "u = np.ones(nx); ## create an array of the length of nx, filled with value 1 m/s\n",
    "u[int(.5 / dx):int(1 / dx + 1)]=2; ## set the values between x=0.5 and x=1 to be 2 m/s\n",
    "print(u)\n",
    "## add your code to plot u here"
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   "source": [
    "Question (c) Now we want to calculate the change of velocity after 25 time steps. You are expected to fill in your codes at location #1 - #5. "
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    "un = np.ones(nx) #initialize a temporary array\n",
    "#1 add for loop for values of n from 0 to nt, so it will run nt times\n",
    "#2 within the for loop, copy the existing values of u into un\n",
    "    #3 loop for advancing the values of u at each point in x\n",
    "    #4 your formula for computing u at current time n+1\n",
    "    #5 plot your solutions at the last time step"
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   "source": [
    "Question (d) Plot and compare velocity profiles at $t=0.625$ when you use $dt = 0.25$ and $dt=0.0025$. You are expected to have three curves on this plot. Comment on the differences. "
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   "source": [
    "Bonus (a), extra 2 points, repeat Questions (a)-(d) using a central difference in space."
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