ODEs arise in many physics problems Classifications: As with the ...

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Adaptive Stepping●●To do adaptive stepping we need:– An error estimate– A method to predict a better timestepError estimate: compare solution with single ¿ step (ys ) to onewith two ¿/2 steps (y d )– Solution with two smaller steps should be more accurate– Relative error is– User supplies a desired error, ²PHY 688: Numerical Methods for (Astro)Physics
Adaptive Stepping●Timestep correction:– Local truncation error of 4 th -order Runge-Kutta is ~ ¿ 5– Estimated timestep to meet our desired accuracy is:– We'll be a little conservative, and use:●●If our step does not meet the desired accuracy, we throw outthe new solution, get the new timestep, and redo the stepLook at the code... and vary tolerancesPHY 688: Numerical Methods for (Astro)Physics
Page 1 and 2: ODEs●●●●ODEs arise in many
Page 3 and 4: Example: Projectile Motion●Air re
Page 5 and 6: Euler's MethodProjectile w/o air-re
Page 7 and 8: Euler-Cromer MethodLet's look at th
Page 9 and 10: Euler Method ConvergenceHere the er
Page 11 and 12: Orbits: Euler's Method●Simplest c
Page 13 and 14: Orbits: Euler's MethodThings get be
Page 15 and 16: Orbits: Euler-Cromer MethodSimple c
Page 17 and 18: Elliptical Orbits●●●We can ju
Page 19 and 20: Higher-order Methods●●Recall th
Page 21 and 22: 2 nd -order Runge-KuttaPHY 688: Num
Page 27 and 28: 4 th -order Runge-KuttaPHY 688: Num
Page 33 and 34: 4 th -order Runge-KuttaEven with a
Page 35 and 36: ConvergenceConvergence using the fi
Page 37 and 38: Adaptive Stepping●●●●We nee
Page 39 and 40: Ex: Highly-Elliptical Orbit●Solut
Page 41: Ex: Highly Elliptical OrbitPHY 688:
Page 45 and 46: Adaptive Stepping●Energy conserva
Page 47 and 48: Adaptive Stepping/Error Estimation
Page 49 and 50: Two-Point Boundary Problems● Rewr
Page 51: Two-Point Boundary ProblemsLook at
Page 54 and 55: Stiff Equations / Implicit Methods
Page 57 and 58: Stiff Equations / Implicit Methods
Page 59 and 60: Stiff Equations / Implicit MethodsN
Page 61 and 62: Stiff Systems of Equations●What a
Page 63 and 64: Stiff Systems of Equations●Backwa
Page 65 and 66: Stiff Systems of EquationsAt 5× th
Page 67 and 68: VODE w/ Linear Stiff SystemVariable
Page 69 and 70: VODE w/ Non-linear Stiff System●E
Page 71 and 72: Practical Definition of Stiff●Com
Page 73: Quick Discussion of Stability●Reg

ODEs arise in many physics problems Classifications: As with the ...

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