# Tag Info

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### 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 ...

### 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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### 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 (...

### 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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### 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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### 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 ...

### 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 ...
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 ...
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 ...
1 vote
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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 ...
1 vote

### LU-SGS and boundary conditions

Check the article in . 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 ...
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 ...
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 ...
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 ...
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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