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Differential Equation 1.3

# Differential Equation 1.3 - Special methods for particle...

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University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei Special methods for particle dynamics Second-order differential equations in which first-order derivatives do not appear are found so frequently in applied problems, particularly those arising from the law of motion, that special methods have been devised for their solution. The idea is to go directly from the second derivatives to the function itself without having to use the first order derivatives. ( ) () t F dt t r d m 2 2 =

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University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei Verlet algorithm for ( ) () t F dt t r d m 2 2 = r(t+h) = -r(t-h) + 2r(t) + h 2 F(t)/m + O(h 4 ) ( ) ( ) ( ) ) h ( O dt t r d 6 h dt t r d 2 h dt t dr h ) t ( r ) h t ( r 4 3 3 3 2 2 2 + + + + = + ( ) ( ) ( ) ) h ( O dt t r d 6 h dt t r d 2 h dt t dr h ) t ( r ) h t ( r 4 3 3 3 2 2 2 + + = Let’s write two third-order Taylor expansions for r(t) at t+h and t-h and sum
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Differential Equation 1.3 - Special methods for particle...

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