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13 votes
Accepted

Why do I get an oscillatory solution when applying the implicit trapezoidal method to the linear diffusion equation?

I think you might have mixed up some terminology. An exponential integrator would use some type of eigensolver or related approach to exactly calculate $$f^{n + 1} = e^{\delta t\cdot \mathcal A}f^n$$ ...
Daniel Shapero'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
6 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

Euler Method Instability. Why?

If we take your ODE, $$\frac{dy}{dx}=-\frac{x^2}{y},$$ multiply both sides by $y$ and integrate up, we see that the solutions look like $$ y^2 = C-\frac{2}{3} x^3. $$ Taking your initial condition, ...
origimbo's user avatar
  • 2,259
4 votes
Accepted

Integrating a nonlinear ordinary differential equation

You don't have just a first-order ODE so you cannot use an explicit Runge-Kutta method. Because of the square term, you cannot bring this into mass matrix form even. Instead, what you have is an ...
Chris Rackauckas's user avatar
4 votes
Accepted

Floating point and global error in Euler Method

In a one-step method $$ y_{n+1}=y_n+h\Phi_f(x_n,y_n,h) $$ one gets a truncation error for the exact solution $$ y(x_{n+1})=y(x_n)+h\Phi_f(x_n,y(x_n),h)+h^{p+1}\tau(x_n) $$ For the error propagation of ...
Lutz Lehmann's user avatar
  • 6,129
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
  • 8,702
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
3 votes

Algorithm to numerically determine whether my computed solution for a 1st order ODE is stable/unstable?

First, you may define a differentiable interpolant for the data produced by the ODE stepper, and then plot the residual $ \Delta(t) := \tilde{y}'(t) - f(\tilde{y},t) $ where $\tilde{y}$ is the ...
user14717's user avatar
  • 2,165
3 votes

Stability region of explicit midpoint method

You have written down the midpoint rule as a two-step method, a member of the family of multi-step methods. For these methods, one can show that a multi-step method $$\alpha_{k} y_{n+k} + \alpha_{k-1} ...
GertVdE's user avatar
  • 6,159
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
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

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
1 vote
Accepted

Adding a diffusion term to the MUSCL - Kurganov and Tadmor central scheme

From a finite volume point of view, fluxes should be calculated at the cell faces and added up for each cell. However, since you are using Cartesian meshes and your material properties are constant, ...
ConvexHull's user avatar
  • 1,443
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
  • 6,129
1 vote
Accepted

DOP853 integration method is missing (SciPy)

I would check what version of scipy you are using. DOP853 was introduced relatively recently in 1.4.0. In 1.4.1, I see DOP853 listed appropriately. ...
Alex's user avatar
  • 241
1 vote
Accepted

Step size and stability of Euler forward method

Using the exact solution to estimate $\lambda$ may not give a time step that works for the numerical scheme. The local estimate of $\lambda$ is $$ \lambda = 400 t u $$ If you try this step $$ h = \min(...
cfdlab's user avatar
  • 3,028
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

Finite Difference Method Limitations/Stability Criteria

In essence it does not matter what $\vec{j}$ means, but what it is. All worth to know is that your conservation equation is something like: $$\frac{\partial m}{\partial t} +\textrm{div}\,\vec{j}(m) = ...
HBR's user avatar
  • 1,648

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