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14 votes
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 ...
helloworld922's user avatar
8 votes
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 ...
Chris Rackauckas's user avatar
7 votes
Accepted

Stiff ODE solver in the web browser

I am the author of diffeq-js (https://github.com/martinjrobins/diffeq-js), which is a javascript library for solving DAE (and ODE) models in the browser using ...
Martin Robinson's user avatar
7 votes
Accepted

Does this second-order implicit Runge-Kutta method have a name?

As Wolfgang stated in the comments, this is not a traditional RK due to the inconsistent time evaluations within a stage. At first it would seem it can't even be cast as an additive RK since terms ...
Steven Roberts's user avatar
7 votes

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 ...
Laurent90's user avatar
  • 1,943
6 votes
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 ...
bgav's user avatar
  • 106
6 votes
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 ...
Chris Rackauckas's user avatar
6 votes
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-...
Kirill's user avatar
  • 11.4k
5 votes
Accepted

Mass matrix and BDF time integration

Your second one is the correct form. BDF approximates derivative with backward difference. Write $$ M(y) \dot{y} = f(y,t) $$ as $$ \dot{y} = M^{-1}(y) f(y,t) =: F(y,t) $$ write the BDF for this $$ \...
cfdlab's user avatar
  • 3,028
5 votes

ODE: should Euler implicit be more accurate than Euler explicit for a given computational step?

In general, just because method A provides certain guarantees (such as unconditional stability, energy conservation, being symplectic) does not imply that it is more accurate. In fact, a common ...
Wolfgang Bangerth's user avatar
5 votes
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 ...
Wolfgang Bangerth's user avatar
4 votes
Accepted

Understanding butcher tableau when it comes to implicit methods

To fix notation, denote the Butcher tableau by $$ \begin{array}{c|c} c & A\\ \hline & b^T \end{array} $$ where $b$ and $c$ are vectors of length $s$ (the number of stages) and $A$ is a $s \...
Will P.'s user avatar
  • 831
4 votes

Finite difference methods

This is exactly the case when the lack of information in the question allows to answer it pretty certainly: it is certainly possible. The error would depend on many factors, including the ...
Anton Menshov's user avatar
  • 8,672
3 votes

Do Explicit Methods Always Require an Analytical Solution

You are confused about ODEs. You think that in order to solve an ODE, you need to know what $f=y'$ is. But it's the other way around: When you try to model something in the real work, you ask "...
Wolfgang Bangerth's user avatar
2 votes
Accepted

From explicit to implicit SSP Runge-Kutta time discretization for DG

You cannot merely adjust an explicit RK scheme into an implicit one, implicit routines are much more involved because each intermediate slope can depend upon slopes 'in the future'. This introduces a (...
cbcoutinho's user avatar
2 votes

For implicit schemes, is there any general result that says numerical diffusion increases with smaller timesteps (for CFL<1) as in explicit schemes?

Short Answer: There is no general result that would hold for all implicit schemes. The reason is that how your method behaves with respect to numerical diffusion depends on the specific combination of ...
Daniel's user avatar
  • 1,273
2 votes
Accepted

Implicit methods for variable coefficients based on equations of state

There are many ways to do this. Your "predictor-corrector" approach is one way. A better conceptual framework is to do a time discretization first and then look at what kind of problem you ...
Wolfgang Bangerth's user avatar
2 votes

How to fill matrix entries for two-dimensional implicit finite-difference for the general case

It really depends on how the matrix will be used. In implicit schemes, you typically solve a system of the form $(I-\gamma A)x=b$, where $\gamma$ is some small number related to a time step. For ...
whpowell96's user avatar
  • 2,478
2 votes
Accepted

Stability Criteria for Numerical Solution of Windkessel Ordinary Differential Equation

Backward Euler and the implicit trapezoidal rule are both unconditionally stable for this problem. If you're seeing instability then you haven't implemented them correctly.
David Ketcheson's user avatar
1 vote

Local truncation error of given implicit 1-step scheme

You insert the exact solution on both sides so that $y'(t_{n+1})=f(t_{n+1},y(t_{n+1}))$ and $y''(t_{n})=f'(t_{n},y(t_{n}))$. Thus \begin{align} O(h^{p+1})=g(h)&=-y(t+h)+y(t)+\frac{h}{6}[4y'(t)+2y'(...
Lutz Lehmann's user avatar
  • 6,109
1 vote

How can I derive a second order implicit method for this coupled ODE update?

In the linear case you get $$k_1+k_2=2My+\frac h2 M(k_1+k_2),$$ so that isolating the sum you get $$k_1+k_2=2(I-hM/2)^{-1}My.$$ For the step this gives $$ y^{n+1}=y^n+\frac h2(k_1+k_2)=y^n+(I-hM/2)^{-...
Lutz Lehmann's user avatar
  • 6,109
1 vote

Finite difference - Explicit / Implicit / Crank Nicolson - Does the implicit method require the least memory?

You seem to have given the 1D equations for the discretizations, even though the problem is in 2D. Regardless, the explicit method requires the least memory since you don't even have to form a ...
Savithru's user avatar
  • 343
1 vote

Solving a PDE implicitly by iteration in python

To answer your questions: Is there any better way to do it without being able to use matrix notation? Yes, there are alternative ways, but this is not the best approach. In other words, this method (...
Hassan Saleem's user avatar
1 vote

Implicit solution to Sylvester equation

Yes, there is active research on that, especially in the Lyapunov case: it turns out under some conditions $M$ is well approximated by a low-rank or banded matrix. So you can find an implicit ...
Federico Poloni's user avatar
1 vote
Accepted

Adding Non-Linear source term to 2d Implicit MATLAB code

You can solve your problem in two ways: explicitly or implicitly. If you want to solve your system explicitly, then the solution is quite simple: the only term you would be solving for is the $u_{i+1}...
cbcoutinho's user avatar
1 vote

Implicit integration for FLIP?

In the computer graphics community there exist many approaches that approximate the advection operator of the Navier-Stokes or Euler equations with unconditional stability by particle tracing that are ...
dweber's user avatar
  • 131
1 vote

How to support or contradict a hypothesis on unconditional stability using numerical optimization

In your expression for $S$, is $x = k_x \Delta x$ and $y = k_y \Delta y$ for some "wave number" vector $\mathbf{k} = [k_x \quad k_y]^T$ with $\Delta x $ and $\Delta y$ being grid spacings in $x$ and $...
hpc's user avatar
  • 305
1 vote

Textbook/Manual on Implicit FEM Methods

The implicit/explicit designation refers to the finite difference time stepping scheme used to compute dynamic models. Nonlinear Finite Element Methods by Peter Wriggers talks about the two different ...
Janus's user avatar
  • 11
1 vote

CFD: Doubt with time convergence in advection fully implicit upwind scheme

You can take a look at the truncation errors in your discretization. For example, for your time discretization, the truncation error would be $$ \frac{\Delta t}{2} \frac{\partial^2 T}{\partial t^2} $$...
GreenEye's user avatar

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