# Tag Info

6

I'm going to write this as an answer although it doesn't directly answer the question. Plugging the second equation and the third equation into the first, and plugging the third into the second, together give: \begin{align} \frac{\partial^2 c}{\partial t^2} &= \frac{\partial^2}{\partial x^2}\frac{\partial c}{\partial t} + \frac{\partial b}{\partial ... 5 I think the work of Kennedy and Carpenter (mentioned already by @GoHokies) is still the definitive study on this topic. The journal paper can be found here; for some reason Google Scholar only provides links to the technical report. It includes methods of up to fifth order that are optimized for a range of properties relevant to convection-diffusion-... 4 Your added edit to the question points exactly the way this needs to be done: You first evolve your field using a half-step with the \hat D operator, then a full step with the \hat N operator, and then another half step with \hat D. These are done one after the other, i.e., the result of one operation is the input for the next. If you're interested in ... 4 What you're proposing is widely referred to as Strang splitting. There is a huge literature on similar methods. There are also many questions on this site on the same topic and a tag for them. Your question seems to suggest that in your case the ODE is linear (but possibly time dependent), and you don't assume any special structure of C. In this case ... 3 Yes, you can use different time steps and time stepping methods for the two sub-problems. In fact, that is one of the reasons why operator splitting methods are so popular: if you had two physical effects -- one slow, and one fast -- then you can use long time steps for the slow process and only need to use small time steps for the fast process. ... 3 Analogous equations are considered in different applications e.g. stationary advection equation with right hand side. Looking to your equation through this view, you should try one-sided finite differences instead of central differences that are known to produce oscillations. For instance if your input data are such that \dfrac{\hbar k}{m} \ge 0 \,, \...

3

The design of the "best" splitting schemes is discussed and investigated at length in this recent paper (disclosure: I am one of its authors). In short, the most commonly used criterion is the size of the leading truncation error term coefficients. This can generally be made smaller by using a larger number of stages, and the tradeoff can be worthwhile if ...

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A new resource for high-order splitting schemes that lists quite a few can be found here: http://www.asc.tuwien.ac.at/~winfried/splitting/

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This look like that it is exactly the same as the first order time splitting scheme except the first and last half time step, and the computation is faster with the reduction. Am I missing something here? You're exactly correct, as others have already said. Also, what is the time splitting method with four order? There are many higher-order splitting ...

2

When solving a linear system $Ax=b$, you suffer from two sources of error: Round-off error The ill-conditionedness of the matrix. The condition number of a computational representation of a matrix is, for all practical purposes equal to the condition number of the exact matrix (which you can't represent computationally), so it does not matter whether you ...

2

You can prove convergence by satisfying the spectral radius relationship you note, choosing $S$ and $T$ such that $\rho(S^{-1}T) < 1$. This comes about by first writing two equations based on your operator splitting: $$Sx = Tx + b$$ $$Sx^{(k+1)} = Tx^{(k)} + b$$ where $x$ is exact solution and $x^{(k)}$ is the $k^{th}$ iteration's solution. Now ...

1

I can describe the simplified version of this algorithm which might help you to understand what's going on here. Let's you have this Time-dependent Schrodinger equation: $$i \hbar \partial_{t} \Psi = -\frac{\hbar^{2}}{2m} \nabla^{2} \Psi$$ For a moment forget about potential cause that complicate things here, which might be necessarily helpful for ...

1

In general, a problem with different parts can be split, then each part is solved separately, then the solutions for each flow are composed back to obtain the final solution. For a description of these methods you can refer to A concise introduction to Geometric Numerical Integration by Blanes & Casas and to Geometric Numerical Integration by Hairer, ...

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Sure, see in Numerical Recipes "Operator Splitting Methods Generally" and follow the references there...

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The Strang splitting method goes like this. You start with the PDE $$u_t = (L+B)u, \qquad L = \partial_x^2,$$ and you notice that when $L$ and $B$ are independent of $x$, the exact solution after time $\delta t$ to this is $$u = e^{(L+B)\delta t}u_0 \approx e^{\frac12 L\delta t}e^{B\delta t}e^{\frac12 L\delta t}u_0.$$ Because $B$ is just a function of $x$...

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You can combine those two steps since both of them involve the same operator ($L_1$). For the fourth-order method I recommend that you check out my answer to a similar question on Stack Overflow and also Section 4 in the following paper (see also the discussion below): Yoshida, H. (1990). Construction of higher order symplectic integrators. Physics Letters ...

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When you write down the product $$M^{-1}N = \begin{pmatrix} m_{11}^{-1} & -m_{11}^{-1} m_{12} m_{22}^{-1} \\ 0 & m_{22}^{-1} \end{pmatrix} \begin{pmatrix} 0 & 0 \\ m_{21} & 0 \end{pmatrix} = \begin{pmatrix} -m_{11}^{-1} m_{12} m_{22}^{-1} m_{21} & 0 \\ m_{22}^{-1} m_{21} & 0 \end{pmatrix}$$ it is clear that the eigenvalues of \$M^{-1}...

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