The forward (explicit) method is unstable for large step sizes. The proof basically just involves some algebra and bounding the growth of the forward method by some function, and then showing that the growth factor results in exponential error if your step size is too large. See: http://www.math.iit.edu/~fass/478578_Chapter_4.pdf. However, by using the backwards method, we no longer have the stability issue, see: https://en.wikipedia.org/wiki/Backward_Euler_method. However, this comes at the cost of increased computational complexity.

justin-shao

I'm having trouble understanding the dynamic error here. I get that we're summing together a term for each time step, but I can't quite understand the two terms being added - I think the first is for the forward euler error and the second is for backward euler error, but I still have a few questions:

First, why is the second term not squared?

Second, what exactly is the semantic meaning of the two terms? Take the first term as example, I thought $q^{t+\Delta t} - q^t$ would be $\Delta t\dot{q}^t$ already. What am I missing here?

Why is backwards more stable?

The forward (explicit) method is unstable for large step sizes. The proof basically just involves some algebra and bounding the growth of the forward method by some function, and then showing that the growth factor results in exponential error if your step size is too large. See: http://www.math.iit.edu/~fass/478578_Chapter_4.pdf. However, by using the backwards method, we no longer have the stability issue, see: https://en.wikipedia.org/wiki/Backward_Euler_method. However, this comes at the cost of increased computational complexity.

I'm having trouble understanding the dynamic error here. I get that we're summing together a term for each time step, but I can't quite understand the two terms being added - I think the first is for the forward euler error and the second is for backward euler error, but I still have a few questions:

First, why is the second term not squared?

Second, what exactly is the semantic meaning of the two terms? Take the first term as example, I thought $q^{t+\Delta t} - q^t$ would be $\Delta t\dot{q}^t$ already. What am I missing here?