Questions tagged [operator-splitting]
For questions on methods for solving partial differential equations by decomposition of a continuous or discrete operator into two or more separate operators.
31 questions
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Implicit-Explicit Operator Splitting Scheme
I am trying to solve the 2D advection-diffusion equation in cylindrical coordinates:
$$
\frac{\partial c}{\partial t} = D\left(\frac{\partial^2 c}{\partial r^2} + \frac{1}{r}\frac{\partial c}{\partial ...
- 11
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140
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Crank-Nicolson vs Spectral Methods for the TDSE
The time-dependent Schroedinger equation (TDSE) depends linearly on the system's initial state $\vert \psi(0) \rangle$, such that the solution can be generally written as
$$ \vert \psi(t) \rangle = \...
- 111
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67
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schrodinger eq time propagation with dissipation using split step operator
I am looking in ways to include energy dissipation while propagating a coherent wavepacket in a 1d TDSE. for example I use the split step method: exp[Δt(D+V)]≈exp[ΔtV/2]exp[ΔtD]exp[ΔtV/2], and ...
- 121
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172
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How does the error work for the Strang Splitting?
We know in Strang splitting that the splitting error in the steady state solution is proportional to $h^2$. I want ask 2 things:
If this error in the steady state solution is the global error?
If we ...
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173
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What is temporal order of accuracy of the PISO algorithm?
A few Computational Fluid Dynamics (CFD) codes implement the so called PISO (Pressure-Implicit with Splitting of Operators) algorithm for pressure-velocity coupling.
My concern is what is actual ...
- 1,413
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1
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230
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How do I apply BDF2 in a STRANG splitting
I have a 3D diffusion equation that I want to solve using a time splitting (2D+1D). Assume that $A$ is the 2D discrete diffusion operator and $B$ is the 1D discrete diffusion operator.
I want to use a ...
- 53
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1
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553
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Operator splitting to solve time dependent Schrödinger equation
I encountered the split operator method to solve the time dependent Schrödinger equation during a lecture. I understand the method on a theoretical basis (I think at least), but I'm struggling to ...
- 131
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3k
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Split-step Fourier method applied on Schrodinger equation
I'm trying to solve a Schrodinger equation of the form $i\frac{\partial}{\partial t}\psi=-\frac{\partial^2}{\partial x^2}\psi + (V(x)+\alpha|\psi|^2)\psi$ using the split-step Fourier method ...
- 21
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90
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Should I expect computational gains using a second-order splitting method here?
I am trying to solve a three-dimensional baroclinic transport problem. The hydrodynamic (three-dimensional shallow water) equations are:
\begin{align}
\nabla \cdot \vec{v} = 0, \tag{1} \\
\frac{\...
- 399
1
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1
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139
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General questions regarding stability for time-integration of operator-split PDE systems
I am interested in solving ODE systems of the form
\begin{align}
\frac{\partial \vec{u}}{\partial t} = F(\vec{u})
\end{align}
where $F$ is a nonlinear operator, $\vec{u}$ is a vector valued function ...
- 399
1
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2
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106
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Operator splitting for 4 subproblems
Typically an ODE System which involves 2 different physical problems such as diffusion and advection can be numerically approached by the well known Strang operator splitting scheme. I'm wondering if ...
- 156
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0
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173
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Split-step-method for coupled equations
I have implemented a split-step-method for an equation of the shape
$$\partial_z E = i\partial_x^2E+ic|E|^2E$$
resulting in a split into the linear part
$$L=\partial_x^2$$
and the nonlinear part
$$N=...
- 563
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0
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158
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How to formulate Poisson's equation into flux eqution
I have a small 2D system I'm trying to model using a non-linear extension of Darcy's law for fluid flow in porous media. I'm primarily interested in the local flow velocity, not necessarily the ...
- 694
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149
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Stability of dark solitons in a harmonic trap
This question is based upon a research article which I am trying to reproduce. One of the main result of this paper is the condition on transverse confinement of the Bose-Einstein Condensate(BEC) to ...
- 135
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0
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78
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Growing error from a smooth initial condition for Fisher KPP equation
I'm studying the Fisker-KPP equation on the line (and in $]0, 100[$ numerically):
$$
\partial_t u = \Delta_{xx} u + u(1-u)
$$
I notice a behavior I don't understand with a smooth initial condition $...
- 443
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2
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131
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Trotter expansions in ode solver?
Trotter expansions say:
$$ e^{A+B} = \lim_{P\to\infty} \big(e^{A/2P} e^{B/P} e^{A/2P} \big)^P. $$
With $P = 2$, it becomes (with high accuracy)
$$ e^{A/4} e^{B/2} e^{A/2} e^{B/2} e^{A/4}. $$
Let's ...
- 179
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1
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1k
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Split operator method
I am new to the field of computational physics and have a couple of questions regarding solving the non-linear Schrödinger equation using Operator splitting.
1) If the hamiltonian is of the form $H=\...
- 135
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2
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81
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Accuracy between ill-conditioned matrix-free vs. matrix-based operators
As far as I know, precision errors become larger as the condition number of a matrix increases.
Consider a matrix-based operator:
$$A = \nabla \bullet k \nabla $$
And a matrix-free operator:
$$\...
- 619
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0
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94
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Direction-splitting for SSP-RK schemes
What are the implications of applying a direction-splitting within each stage of an SSP-RK scheme?
For instance, given a standard advective transport type equation:
$$
\partial_{t}Q + \operatorname{...
- 534
2
votes
1
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121
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High order time splitting methods
There are lots of higher order time splitting method as shown by the list with real and complex coefficients $a_i, b_i, c_i$:
$$
[e^{c_s \Delta t \hat C}] e^{b_s \Delta t \hat B} e^{a_s \Delta t \...
- 198
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Strang splitting
I have recently come across the Strang splitting and have some questions. For the differential equation of the form
$$ dy/dt = (L_1 + L_2)y$$
Strang splitting implement the time splitting as
$$
\...
- 198
3
votes
1
answer
432
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Convergence conditions of a stationary iteration method for linear systems
Recently, I obtain a linear system, $Ax = b$, where $A$ is a nonsingular, strictly diagonally dominant $M$-matrix. Then I also got a matrix splitting $A = S - T$, where $S$ is also a nonsingular, ...
- 129
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220
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Evolving nonlinear Schrodinger equation with higher-order algorithms?
First I will give the relevant information for my question, and then I'll ask the question.
$\large{\textrm{Background}}$
For evolving the nonlinear Schrodinger equation (NLS), one typically uses [a ...
- 489
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votes
1
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323
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Stability in discretization of a PDE
Suppose I want to numerically solve for $f(x,k)$ the one-dimensional Boltzmann equation for electrons in steady-state condition, given as:
\begin{equation}
\left( \dfrac{\hbar k}{m} \right) \dfrac{\...
- 430
7
votes
1
answer
738
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Fourth order IMEX Runge-Kutta method
I am looking for the Butcher tableau of a fourth order accurate Runge-Kutta method with IMEX splitting. I have been reading the ''classical'' paper on the subject by Ascher, Ruuth and Spiteri as well ...
- 1,273
2
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0
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222
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Segregated solving of a tightly coupled system of PDEs
To compute the evolution of a free surface between two incompressible, immiscible liquids, two tightly coupled equations have to be solved, the volume fraction advection and the Navier-Stokes ...
- 729
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169
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Derivation of a Higher Order Compact Alternating Direction Implicit Method
I dont understand how this Higher Order Compact ADI scheme, which is fourth order in time and space, for the wave equation is derived:
I go through the following
Using Tylor's expansion $u(t+h,x,y)$ ...
- 11
14
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0
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458
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Operator Splitting methods for DAEs
After doing some research, I've found that most of the literature on operator splitting methods (e.g. Strang Splitting, Fractional Step, etc.) are specifically designed for a standard problem type of ...
- 12.2k
3
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1
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174
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Iterative Block Matrix Splitting for Multiphysics Simulation
I have a problem of the form
$$\left[\begin{array}{cc}
-(\lambda+2\mu)\frac{d^2}{dx^2} & \alpha\frac{d}{dx} \\
\frac{\alpha}{\Delta t}\frac{d}{dx} & \frac{c_0}{\Delta t}I-\frac{\kappa}{\...
- 12.2k
9
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1
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1k
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Optimal use of Strang splitting (for reaction diffusion equation)
I made a strange observation while computing the solution to a simple 1D reaction diffusion equation:
$\frac{\partial}{\partial t}a=\frac{\partial^2}{\partial x^2}a-ab$
$\frac{\partial}{\...
- 2,161
17
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5
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639
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Are there operator splitting approaches for multiphysics PDEs that achieve high order convergence?
Given an evolution PDE
$$u_t = Au + Bu$$
where $A,B$ are (possibly nonlinear) differential operators that don't commute, a common numerical approach is to alternate between solving
$$u_t = Au$$
...
- 16.9k