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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4
votes
1answer
34 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 ...
3
votes
1answer
83 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) ...
4
votes
1answer
98 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 ...
1
vote
0answers
87 views

Splitting Operator

I have a problem with this finite element formulation. After applied a Splitting Operator $Q=\hat{Q} + \tilde{Q}$ I do not know how to procede. I need to obtain the solution of the following finite ...
2
votes
0answers
127 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 ...
1
vote
0answers
134 views

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 $u(t+h,x,y)$ ...
10
votes
0answers
223 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 ...
3
votes
1answer
113 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 ...
9
votes
1answer
656 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: $\frac{\partial}{\partial t}a=\frac{\partial^2}{\partial x^2}a-ab$ ...
12
votes
4answers
377 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 $$u_t = Au$$ ...