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Ordinary Differential Equations. Jyun-Ming Chen. Review Euler’s method 2 nd order methods Midpoint Heun’s Runge-Kutta Method. Systems of ODE Stability Issue Implicit Methods Adaptive Stepsize. Contents. Review. DE (Differential Equation)
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Ordinary Differential Equations Jyun-Ming Chen
Review Euler’s method 2nd order methods Midpoint Heun’s Runge-Kutta Method Systems of ODE Stability Issue Implicit Methods Adaptive Stepsize Contents
Review • DE (Differential Equation) • An equation specifying the relations among the rate change (derivatives) of variables • ODE (Ordinary DE) vs. PDE (Partial DE) • The number of independent variables involved
Solution of an equation: Solution of DE vs. Solution of Equation f(x) x Review (cont) • Geometrically,
Solution of an differential equation: Geometrically: x t Need additional conditions to specify a solution Review (cont)
Review (cont) • Order of an ODE • The highest derivative in the equation • nth order ODE requires n conditions to specify the solution • IVP (initial value problem): All conditions specified at the same (initial) point • BVP (boundary value problem): otherwise
IVP vs. BVP Physical meaning
Ode2: solves 1st and 2nd order ODE Ic1, ic2, bc: setting conditions ‘ do not evaluate Maxima on ODE
Linear ODE • Linearity: • Involves no product nor nonlinear functions of y and its derivatives • nth order linear ODE
Focus of This Chapter • Solve IVP of nth order ODE numerically • e.g.,
ODE (IVP) • First order ODE (canonical form) • Every nth order ODE can be converted to n first order ODEs in the following method:
The Canonical Problem This is Euler’s method
Example Compare with exact sol:
y 1 y=e–x x Example (cont)
Error Analysis(Geometric Interpretation) Think in terms of Taylor’s expansion If the true solution were a straight line, then Euler is exact
Error Analysis(From Taylor’s Expansion) Euler’s Euler’s truncation error O(Dx2) per step 1st order method
y Cumulative Error x Remark: Dx Error But computation time x = 0 x = T Number of steps = T/Dx Cumulative Err. = (T/Dx) O(Dx2) = O(Dx)
Methods Improving Euler Motivated by Geometric Interpretation
Example (Heun’s) Note the result is the same as Midpoint!?
Comparison of Euler, Heun, midpoint 1st order: Euler 2nd order: Heun, midpoint “order”: All are special cases of RK (Runge-Kutta) methods Remark
RK 4th Order • Mostly commonly used one • Higher order … more evaluation, but less gain on accuracy Classical 4th order RK
System of ODE • Convert higher order ODE to 1st order ODEs • All methods equally apply, in vector form
Initial Condition c m k x Example (Mass-Spring-Damper System) • Governing Equation • After setting the initial conditions x(0) and x’(0), compute the position and velocity of the mass for any t > 0
Assume m=1,c=1, k=1 (for ease of computation) Example (cont) set Dt=0.1
Symptom: Unstable Spring System Become unstable instantly … Start with this … Cause by stiff (k=4000) springs
Example Problem: Stability (cont) Conditionally stable
Discussion • Different algorithm different stability limit • Check Midpoint Method • Different problem different stability limit • use the previous problem as benchmark
Review: Numerical Differentiation Taylor’s expansion: Forward difference Backward difference
Numerical Difference (cont) Central difference
Implicit Method (Backward Euler) Forward difference Backward difference
Example • Remark: • Always stable (for this problem) • Truncation error the same as Euler (only improve the stability)
Stability limit Stiff Set of ODE Use the change of variable Get the following solution: A stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small