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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.

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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 ...
mht's user avatar
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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 = \...
QuantumBrick's user avatar
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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 ...
yourds's user avatar
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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 ...
Giannis Kavroulakis's user avatar
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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 ...
Johntra Volta's user avatar
2 votes
1 answer
223 views

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 ...
Chack.Flack's user avatar
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1 answer
495 views

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 ...
Sito's user avatar
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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 ...
decarat's user avatar
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3 votes
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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{\...
A. B. Marnie's user avatar
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1 answer
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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 ...
A. B. Marnie's user avatar
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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 ...
OD IUM's user avatar
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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=...
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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 ...
cbcoutinho's user avatar
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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 ...
Abhijit's user avatar
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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 $...
bela83's user avatar
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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 ...
Ka Wa Yip's user avatar
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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=\...
Abhijit's user avatar
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2 answers
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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: $$\...
Charles's user avatar
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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{...
Darren Engwirda's user avatar
2 votes
1 answer
115 views

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 \...
unsym's user avatar
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2 answers
2k views

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 $$ \...
unsym's user avatar
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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, ...
Hsien-Ming Ku's user avatar
5 votes
1 answer
211 views

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 ...
Arturo don Juan's user avatar
2 votes
1 answer
315 views

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{\...
Maziar Noei's user avatar
7 votes
1 answer
697 views

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 ...
Daniel's user avatar
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2 votes
0 answers
217 views

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 ...
akid's user avatar
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1 vote
0 answers
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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 $u(t+h,x,y)$ ...
Yahya's user avatar
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14 votes
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452 views

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 ...
Paul's user avatar
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3 votes
1 answer
171 views

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}{\...
Paul's user avatar
  • 12k
9 votes
1 answer
1k views

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}{\...
Thomas Klimpel's user avatar
17 votes
5 answers
636 views

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 $$u_t = Au$$ ...
David Ketcheson's user avatar