Time-Reversal Symmetry and Symmetric Numerical Methods

2 Time-Reversal Symmetry and Symmetric Numerical Methods 

The flow map gives a solution of the differential equation, thus for a time- 

From this we see that if we start from a pointzo and integrate forward in time (i.e apply then apply the involution R, we get the exact same result as if we start fromz(0) = Rzo and integrate backward in time (i.e apply -0- In sum, since zo is an arbitrary point, 

We may seek numerical methods that mimic this property, which we refer to time-reversible numerical methods. Recall from the previous chapter that a numerical method % was said to be symmetric if it satisfied 

It does not automatically follow that a symmetric numerical method is time-reversible, i.e. satisfies a relation inspired by (3.14), namely 

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