\n",
"**Figure Weather Balloon:** A weather balloon oscillating about its level of neutral\n",
"buoyancy.\n",
"
"
]
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"#### Problem One\n",
"\n",
"\n",
"a\\) Using the centered difference formula ([Centered Second Derivative](#lab1:eq:centered-diff2)) for the second derivative, and\n",
" the forward difference formula ([Forward Difference Formula](#lab1:eq:forward-diff')) for the first derivative at the point\n",
" $t_i$, derive a difference scheme for $y_{i+1}$, the vertical\n",
" displacement of the weather balloon.\n",
"\n",
"b\\) What is the difference between this scheme and the forward Euler\n",
" scheme from [Example Seven](#Example-Seven] , *Fix Link* related to the initial\n",
" conditions? (**Hint:** think about starting values …)\n",
"\n",
"c\\) Given the initial values above, explain how to start the numerical\n",
" integration.\n",
" \n",
"*Note*: There are a number of problems in the text of each lab. See the syllabus for which problems you are assigned as part of your course. That is, you don't have to do them all!"
]
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"### Systems of First-order ODE's\n",
"\n",
"\n",
"Discretization extends in a simple way to first-order systems of ODE’s,\n",
"which arise in many problems, as we will see in some of the later labs.\n",
"For now, though, we can see:\n",
"\n",
"#### Example 9\n",
"\n",
"\n",
"The second order DE for the weather balloon problem from\n",
"Example [Eight](#Example-Eight) can be rewritten by letting $u=dy/dt$. Then,\n",
"\n",
"\\begin{align}\n",
"\\frac{dy}{dt} &= u\\\\\n",
"\\frac{du}{dt} &= -\\frac{\\beta}{m} u - \\frac{\\gamma}{m} y\n",
"\\end{align}\n",
"\n",
"which is a\n",
"system of first order ODE’s in $u$ and $y$. This set of differential\n",
"equations can be discretized to obtain another numerical scheme for the\n",
"weather balloon problem."
]
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"#### Problem Two\n",
"\n",
"\n",
"a\\) Derive a difference scheme for the problem based on the above system\n",
" of two ODE’s using the forward difference formula for the first\n",
" derivative.\n",
"\n",
"b\\) By combining the discretized equations into one equation for y, show\n",
" that the difference between this scheme and the scheme obtained in\n",
" problem one is the difference formula for the second derivative."
]
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"### Boundary Value Problem\n",
"\n",
"[lab1:sec:bvp]: (#6.3-Boundary-Value-Problems)\n",
"\n",
"\n",
"So far, we’ve been dealing with *initial value problems* or *IVP’s*\n",
"(such as the problem of heat conduction in a rock in\n",
"Example [One](#Example-One)): a differential equation is given for an\n",
"unknown function, along with its initial value. There is another class\n",
"of problems, called "
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"Image(filename='pdes/pde-grid.png', width='40%')"
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"\n",
" **Figure PDE Grid:** The computational grid for the heat diffusion problem, with\n",
" discrete points $(x_i,t_n)$.\n",
"
\n",
"\n",
"2) Replace the dependent variables (in this example, just the\n",
" temperature $u(x,t)$) with approximations defined at the grid\n",
" points: $$U_i^n \\approx u(x_i,t_n).$$ The boundary and initial\n",
" values for the discrete temperatures can then be written in terms of\n",
" the given information.\n",
"\n",
"3) Approximate all of the derivatives appearing in the problem with\n",
" finite difference approximations. If we use the centered difference\n",
" approximation ([Centered Second Derivative](#lab1:eq:centered-diff2)) for the second derivative in $x$, and\n",
" the forward difference formula ([Forward Difference Formula](#lab1:eq:forward-diff')) for the time derivative (while evaluating the\n",
" terms on the right hand side at the previous time level), we obtain\n",
" the following numerical scheme:\n",
"\\begin{equation} \n",
" U_i^{n+1} = U_i^n + \\frac{\\alpha^2 \\Delta t}{(\\Delta x)^2} \\left(\n",
" U_{i+1}^n - 2 U_i^n + U_{i-1}^n \\right)\n",
"\\end{equation}\n",
"\n",
"Given the initial values, $U_i^0=u_0(x_i)$, and boundary values\n",
"$U_0^n=U_M^n=0$, this difference formula allows us to compute values of\n",
"temperature at any time, based on values at the previous time.\n",
"\n",
"There are, of course, other ways of discretizing this problem, but the\n",
"above is one of the simplest."
]
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"## Mathematical Notes\n",
"\n",
"\n",
"\n",
"### Solution to the Heat Conduction Equation\n",
"\n",
"\n",
"\n",
"In Example [One](#Example-One), we had the equation\n",
"$$\\frac{dT}{dt} = -\\lambda (T-T_a),$$ subject to the initial condition\n",
"$T(0)$. This equation can be solved by *separation of variables*,\n",
"whereby all expressions involving the independent variable $t$ are moved\n",
"to the right hand side, and all those involving the dependent variable\n",
"$T$ are moved to the left $$\\frac{dT}{T-T_a} = -\\lambda dt.$$ The\n",
"resulting expression is integrated from time $0$ to $t$\n",
"$$\\int_{T(0)}^{T(t)} \\frac{dS}{S-T_a} = -\\int_0^t\\lambda ds,$$ (where\n",
"$s$ and $S$ are dummy variables of integration), which then leads to the\n",
"relationship $$\\ln \\left( T(t)-T_a)-\\ln(T(0)-T_a \\right) = -\\lambda t,$$\n",
"or, after exponentiating both sides and rearranging,\n",
"$$T(t) = T_a + (T(0)-T_a)e^{-\\lambda t},$$ which is exactly the [Conduction Solution](#lab1:eq:conduction-soln) equation."
]
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"## References\n",
"\n",
"\n",
"\n",
"Boyce, W. E. and R. C. DiPrima, 1986: Elementary Differential Equations and Boundary Value Problems. John Wiley & Sons, New York, NY, 4th edition.\n",
"
\n",
"\n",
"Burden, R. L. and J. D. Faires, 1981: Numerical Analysis. PWS-Kent, Boston, 4th edition.\n",
"
\n",
"\n",
"Chen, J.-P., 1994: Predictions of saturation ratio for cloud microphysical models. Journal of the Atmospheric\n",
"Sciences, 51(10), 1332–1338.\n",
"
\n",
"Garcia, A. L., 1994: Numerical Methods for Physics. Prentice-Hall, Englewood Cliffs, NJ.\n",
"
\n",
"Strang, G., 1986: Introduction to Applied Mathematics. Wellesley-Cambridge Press, Wellesley, MA.\n",
"
"
]
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"## Glossary\n",
"\n",
"\n",
"**backward difference discretization:** used to estimate a derivative – uses the current points and points with smaller independent variable.\n",
"\n",
"**boundary value problem:** a differential equation (or set of differential equations) along with boundary values for the unknown functions. Abbreviated BVP.\n",
"\n",
"**BVP:** see *boundary value problem*\n",
"\n",
"**centre difference discretization:** used to estimate a derivative – uses a discretization symmetric (in\n",
"independent variable) around the current point.\n",
"\n",
"**closed domain:** a domain for which the value of the dependent variables is known on the boundary of the domain.\n",
"\n",
"**converge:** as the discretization step (eg. ∆t) is reduced the solutions generated approach one solution curve.\n",
"\n",
"**DE:** see *differential equation*\n",
"\n",
"**dependent variable:** a variable which is a (possibly unknown) function of the independent variables in a problem; for example, in a fluid the pressure can be thought of as a dependent variable, which depends on the time t and position (x, y, z).\n",
"\n",
"**differential equation:** an equation involving derivatives. Abbreviated DE.\n",
"\n",
"**discretization:** when referring to DE’s, it is the process whereby the independent variables are replaced by a *grid* of discrete points; the dependent variables are replaced by approximations at the grid points; and the derivatives appearing in the problem are replaced by a *finite difference* approximation. The discretization process replaces the DE (or DE’s) with an algebraic equation or finite system of algebraic equations which can be solved on a computer.\n",
"\n",
"**finite difference:** an approximation of the derivative of a function by a difference quotient involving values of the function at discrete points. The simplest method of deriving finite difference formulae is using Taylor series.\n",
"\n",
"**first order differential equation:** a differential equation involving only first derivatives of the unknown functions.\n",
"\n",
"**forward difference discretization:** used to calculate a derivative – uses the current points and points with larger independent variable.\n",
"\n",
"**grid:** when referring to discretization of a DE, a grid is a set of discrete values of the independent variables, defining a *mesh* or array of points, at which the solution is approximated.\n",
"\n",
"**independent variable:** a variable that does not depend on other quantities (typical examples are time, position, etc.)\n",
"\n",
"**initial value problem:** a differential equation (or set of differential equations) along with initial values for the unknown functions. Abbreviated IVP.\n",
"\n",
"**interpolation:** a method for estimating the value of a function at points intermediate to those where its values are known.\n",
"\n",
"**IVP:** initial value problem\n",
"\n",
"**linear:** pertaining to a function or expression in which the quantities appear in a linear combination. If $x_i$ are the variable quantities, and $c_i$ are constants, then any linear function of the $x_i$ can be written in the form $c_0 + \\sum_i c_i \\cdot x_i$.\n",
"\n",
"**linear interpolation:** interpolation using straight lines between the known points\n",
"\n",
"**Navier-Stokes equations:** the system of non-linear PDE’s that describe the time evolution of the flow of\n",
"a fluid.\n",
"\n",
"**non-linear:** pertaining to a function or expression in which the quantities appear in a non-linear combination.\n",
"\n",
"**numerical instability:** although the continuous differential equation has a finite solution, the numerical solution grows without bound as the numerical interation proceeds.\n",
"\n",
"**ODE:** see *ordinary differential equation*\n",
"\n",
"**open domain:** a domain for which the value of one or more dependent variables is unknown on a portion\n",
"of the boundary of the domain or a domain for which one boundary (say time very large) is not specified.\n",
"\n",
"**ordinary differential equation:** a differential equation where the derivatives appear only with respect to one independent variable. Abbreviated ODE.\n",
"\n",
"**partial differential equation:** a differential equation where derivatives appear with respect to more than one independent variable. Abbreviated PDE.\n",
"\n",
"**PDE:** see *partial differential equation*\n",
"\n",
"**second order differential equation:** a differential equation involving only first and second derivatives of the unknown functions.\n",
"\n",
"**separation of variables:** a technique whereby a function with several dependent variables is written as a product of several functions, each of which depends on only one of the dependent variables. For example, a function of three unknowns, u(x, y, t), might be written as u(x, y, t) = X(x) · Y (y) · T (t)."
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