# Tag Info

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 $$f^{n + 1} = e^{\delta t\cdot \mathcal A}f^n$$ ...
• 10.4k
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 ...
• 12.4k

### 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 ...
• 55.9k
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, ...
• 2,259
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 ...
• 12.4k
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 ...
• 6,129

### 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 ...
• 8,702

### 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 "...
• 55.9k

### 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 ...
• 2,165

• 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 ...
• 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) = ...
• 1,648

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