13 votes
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Non-conservative implementation implicit Euler

This might seem extreme, but this can be analysed exactly. Take the system $$ \dot x_1 = x_2, \qquad \dot x_2=-x_1, \qquad x_1(0) = 1, \qquad x_2(0)=0. $$ Let $X=(x_1,x_2)$ be the state vector, $\...
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  • 11.4k
8 votes
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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 ...
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  • 2,961
8 votes
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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 ...
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8 votes
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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 ...
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6 votes
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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 ...
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  • 96
6 votes
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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 ...
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6 votes
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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-...
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5 votes
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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 ...
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5 votes
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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 $$ \...
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  • 2,983
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 ...
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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 ...
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  • 8,362
4 votes
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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 \...
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  • 611
3 votes

Why is Crank-Nicolson considered implicit in time?

The Crank-Nicolson method is: $\frac{u^{n+1}_{i}-u^{n}_{i}}{dt} = \frac{1}{2}(F^{n+1}_{i}+F^{n}_{i})$ This method calculates the next state of the system, i.e. $u^{n+1}_{i}$, by solving an equation ...
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  • 1,859
2 votes

Not getting correct numerical solution for Advection-Diffusion-Reaction eqn

One of the best ways to test a PDE solver is to use the method of manufactured solutions. Essentially, you modify the PDE (and discretization) by adding a source term that yields an exact solution ...
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2 votes
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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 (...
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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 ...
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  • 1,097
2 votes
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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.
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2 votes
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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 ...
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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 ...
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  • 1,198
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 ...
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  • 323
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 ...
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1 vote
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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}...
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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 ...
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  • 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 $...
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  • 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 ...
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  • 11
1 vote

LU-SGS and boundary conditions

Check the article in [1]. The $\delta Q_f$ (for a boundary face) must be transformed by a matrix $T$ such that, $\delta Q_f = T * \delta Q_L$ ( note $L$ is the left cell and for a boundary face ...
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  • 11
1 vote
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Implicit ODE solver with discontinuous derivatives

With the discussion clearing up some of the confusion in the original post, here is a summary of the discussion so far: I want to implement an implicit ODE solver, but don't know what to do when ...
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1 vote

ADR equation implicit solution: Penta-diagonal matrix for a 2D $N\times N$ system

This is mostly just a matter of assigning an index to each vertex. If you have $n\times n$ unknowns $C_{i,j}$, $0\leq i,j\leq n-1$, then you would typically map the variable corresponding to the ...
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  • 11.4k
1 vote

Spectral Coefficients of Implicit Finite Difference Solution

It depends on how well resolving your mesh is. On some level of coarseness you will first observe $2h$ waves (smallest resolvable waves on your mesh), and soon after, your solution will blow-up. That ...
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1 vote

Initializing implicit linear multistep methods

Just to add a bit to @Jesse Chan's answer: you can preserve 6th-order convergence if you use a 5th-order starting method; in general, the starting method can be one order lower than the multistep ...
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