# Tag Info

Accepted

### Are stiffness and instability equivalent?

There are non-stiff problems which are unconditionally unstable with some explicit methods, and conversely there are stiff problems which can be stable with explicit methods. Consider the oscillating ...
• 1,751
Accepted

### Initializing implicit linear multistep methods

The standard approach is to use a self-starting time-marching algorithm with sufficiently small timestep (such that the order of accuracy is not spoiled) and compute the 5 non-initial value previous ...
• 3,062
Accepted

### Why is Crank-Nicolson considered implicit in time?

A simplification - the Crank-Nicolson method uses the average of the forward and backward Euler methods. The backward Euler method is implicit, so Crank-Nicolson, having this as one of its ...
• 236
Accepted

### Solving PDE implicitly or explicitly depending on stiffness

If you just slap together an implicit and an explicit method you will likely have order loss. You can do so with low order methods though, and Crank-Nicholson mixed with some other integrator is an ...
Accepted

### GMRES vs Newton-GMRES for Solving nonlinear PDE's

The reason is that GMRES can only be used for solving linear equations, i.e. equations of the form $Ax=b$, where $A$ is some matrix and $x,b$ are vectors. What GMRES does, essentially, is it ...
• 106
Accepted

### Textbook/Manual on Implicit FEM Methods

I think you're mixing up ideas. The best thing here is to know the foundations, and it then "implicit FEM" will be a trivial idea (which is why there won't be a book specifically about that). Finite ...
Accepted

### Solve an ODE with positivity-preserving property unconditionally

The two properties are usually called positivity-preserving and monotonicity-preserving (makes it easier to find this question). Looking at http://www.ams.org/journals/mcom/2006-75-254/S0025-5718-05-...
• 11.4k

### Are stiffness and instability equivalent?

I'd like to add a few complements to the accepted answer. Some problems possessing some eigenvalues with positive real parts have "physically" unstable modes, which may actually be damped by ...
• 1,669
Accepted

### Why is this method for simulating a system of springs and masses unstable?

The implicit Euler method is unconditionally stable alright, but what you are doing is not the implicit Euler method. Rather, what you do is compute where the particle would be at the end of the time ...
• 52.4k