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Questions tagged [explicit-methods]

For questions about explicit differential equation algorithms, that directly relate the next time step of some variable y to some function of y at the current time step.

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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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0 answers
50 views

Ways to alleviate my CFL restriction?

I have an advection problem with a sever time stepping restriction. I am using explicit RK type time stepping schemes and standard finite difference so everything is matrix free. Other than going for ...
CuteCompute's user avatar
1 vote
1 answer
97 views

Adding a diffusion term to the MUSCL - Kurganov and Tadmor central scheme

Im currently using a MUSCL scheme with a rusanov flux and Van Leer limiter to simulate the 2d euler equations: $$ \frac{\partial \rho}{\partial t} + \frac{\partial \rho v_x}{\partial x} + \frac{\...
user46777's user avatar
0 votes
1 answer
64 views

Local truncation error of given implicit 1-step scheme

I'm given the 1-step implicit scheme $$y_{n+1} = y_n + \frac{h}{6}[4f(t_n, y_n) + 2f(t_{n+1}, y_{n+1}) + hf'(t_n, y_n)],$$ where $y'(t) = f(t, y)$, and I'm seeking the scheme's local truncation error. ...
jackyooo's user avatar
0 votes
0 answers
40 views

First-order modified Patankar–Euler scheme (MPE)

Is the first-order Modified Patankar–Euler scheme (MPE) an implicit or explicit method? Is there an open-source code implementing the MPE scheme for a system of ODEs?
Mahmoud Saleh's user avatar
3 votes
0 answers
203 views

Python code of explicit method of a nonlinear a BVP

I am trying to have a Python code for the following nonlinear BVP: $$\frac{\partial N}{\partial t}=\frac{\partial^2 N}{\partial x^2}+N(1-N)-\sigma N$$ $$N(0,x)=\sin(2\pi x)$$ $$N(t,0)=0 \hspace{3mm}N(...
Peachy April's user avatar
4 votes
1 answer
586 views

Why do I get an oscillatory solution when applying the implicit trapezoidal method to the linear diffusion equation?

I wish to solve the following equation, $$\frac{\partial f}{\partial t}=\frac{\partial}{\partial x}\left(D(x)\frac{\partial f}{\partial x}\right)$$ using an exponential integrator. I discretize this ...
Sayan's user avatar
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1 vote
1 answer
118 views

Do Explicit Methods Always Require an Analytical Solution

Following some comments from another question I wanted to ask: does an explicit method always require some sort of analytical function/solution? Let's take Euler for example. You have a function $f$ ...
cgbsu's user avatar
  • 33
2 votes
1 answer
62 views

Algorithm to numerically determine whether my computed solution for a 1st order ODE is stable/unstable?

We were given an assignment where we had to determine the numerical solution of Dahlquist's equation $\dot x$ = $\lambda x$, ($\lambda$ = $-7$) for time steps ${0.5,0.25,0.125}$ using explicit euler ...
KARTIK BALI's user avatar
3 votes
1 answer
574 views

For implicit schemes, is there any general result that says numerical diffusion increases with smaller timesteps (for CFL<1) as in explicit schemes?

For the first-order explicit upwind scheme, it can be easily shown that, if one keeps the same grid size and progressively decreases the time step below the max allowed one (i.e. below CFL~1) the ...
Millemila's user avatar
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5 votes
1 answer
413 views

Floating point and global error in Euler Method

Inspired by this answer, I tried to understand when floating point errors come into visibility and to check it also comparing the plot of the numerical solution with Explicit Euler with the analytical ...
Vefhug's user avatar
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1 vote
1 answer
963 views

DOP853 integration method is missing (SciPy)

I was checking some integration methods provided by SciPy, in which the DOP853 should be included according to the webpage (https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate....
user avatar
2 votes
1 answer
2k views

Step size and stability of Euler forward method

I'm trying to calculate the maximum step size that provides stability for the following nonlinear IVP using the Euler forward method: $u'(t) = -200tu(t)^2,\qquad u_0 = 1, \qquad t\in [0,3]$, with ...
Joscha Fregin's user avatar
1 vote
1 answer
126 views

Finite difference methods

I am currently applying the finite difference method to the solution of the diffusion equation. I think that a problem has occurred, and is as follows, my explicit method is the most accurate when ...
Peter's user avatar
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3 votes
1 answer
2k views

Stability region of explicit midpoint method

Consider the explicit midpoint method, i.e $$y_{n+1}-y_{n-1} = 2hf(y_n).$$ I'm asked to apply this method to the linear test equation, $f(y_n) = \lambda y_n,$ in order to find the method's stability ...
Heuristics's user avatar
1 vote
1 answer
311 views

Finite difference - Explicit / Implicit / Crank Nicolson - Does the implicit method require the least memory?

Examine a dynamic 2D heat equation $\dot{u} = \Delta u$ with zero boundary temperature. A standard finite difference approach is used on a rectangle using a $n\times n$ grid. For the resulting linear ...
user29635's user avatar
3 votes
1 answer
1k views

ODE: should Euler implicit be more accurate than Euler explicit for a given computational step?

I am aware than Euler explicit is conditionally stable, and Euler implicit is unconditionally stable. And I am aware that it is probably pointless to use Euler implicit with a small computational step ...
Millemila's user avatar
  • 435
2 votes
1 answer
180 views

Euler Method Instability. Why?

I am currently enjoying writing computational codes as a hobby. Right now I've worked out an Euler method and results are pretty good with up to $x=1$. Over $x=1$, instability starts to kick in. May I ...
Kelvin Loh's user avatar
1 vote
1 answer
162 views

Stability Criteria for Numerical Solution of Windkessel Ordinary Differential Equation

I'm trying to solve this equation (Windkessel equation) numerically as: $$C \frac{d P}{d t} + \frac{P}{R} = Q(t)$$ Where $C$ is compliance, $R$ is resistance, $P$ is pressure, and $Q(t)$ is a known ...
GGG's user avatar
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5 votes
1 answer
445 views

Solving PDE implicitly or explicitly depending on stiffness

I've got a system of several PDEs for a multitude of parts which represent real hydraulic parts like pipes or thermal energy storages. Each of these parts may have an arbitrary number of nodes and/or ...
JE_Muc's user avatar
  • 265
1 vote
1 answer
74 views

Integrating a nonlinear ordinary differential equation

I am solving an equation of the form $(*)$ $0 = a(f) (\partial_rf)^2 + b(f) (\partial_rf) + c(f),$ where $f$ is a real function of $r\in \mathbb{R}$, and $a,b,c$ are real functions of $f$. The ...
physics_researcher's user avatar
0 votes
1 answer
1k views

Finite Difference Method Limitations/Stability Criteria

Is it possible to solve an equation with only a single derivative such as: $$\frac{\partial U(x,t)}{\partial t} = A - BU(x,t)$$ with finite difference methods? I ask as I am trying to solve the ...
Matt Ellis's user avatar
0 votes
1 answer
193 views

From explicit to implicit SSP Runge-Kutta time discretization for DG

In Hesthaven book (Nodal Discontinuous Galerkin Methods) he uses SSP Runge-Kutta time method which is explicit. How can I change the explicit RK to an implicit one?
Geo's user avatar
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1 vote
0 answers
210 views

Stability of the explicit MacCormack Scheme to solve the Navier Stokes equations with Wilcox's K-Omega Turbulence Model

I am solving turbulent pipe flow with an explicit MacCormack scheme and Wilcox k-omega model. The laminar version of the code had three distinct stability criteria which worked fine after ...
Ferferimori's user avatar
2 votes
0 answers
421 views

Collocated Grid Navier Stokes Solver

I want to solve Navier Stokes equations on a collocated grid. Earlier, I was using a MacCormick scheme based solver where I discretized predictor step in forward differences and corrector step in ...
Tanmay Agrawal's user avatar
0 votes
1 answer
1k views

The condition for stability using the leapfrog method

I have the ODE below $$\frac{d}{dt}\pmatrix{x\\ y} = \pmatrix{0 &1\\-a &0}\pmatrix{x\\ y} \enspace .$$ The $m=1$ leapfrog method is defined as: $$y_{n+1} = y_{n-1} + 2f_nh \enspace .$$ For ...
SomePhysicsStudent's user avatar
1 vote
0 answers
42 views

Resources for viscous behavior in simple FEM

I am working on a simple explicit-integration lumped-mass elastic FEM code which implements CST+DKT triangles (plate+shell) and constant-strain tetrahedra (http://woodem.eu/doc/theory/membrane-element....
eudoxos's user avatar
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1 vote
2 answers
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Performance metrics to compare initial-boundary value problem solutions

I am comparing the performance several finite difference methods of solving an initial-boundary value problem. There are several dimensions to this comparison: Number of cells Number of timesteps ...
Neal Kruis's user avatar
16 votes
2 answers
3k views

Is it possible to solve nonlinear PDEs without using Newton-Raphson iteration?

I am trying to understand some results and would appreciate some general comments on tackling nonlinear problems. Fisher's equation (a nonlinear reaction-diffusion PDE), $$ u_t = du_{xx} + \beta u ...
boyfarrell's user avatar
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10 votes
3 answers
8k views

What is the difference between implicit FEM and explicit FEM?

What is the difference between explicit FEM and implicit FEM exactly? According to the post here, it seems that the only difference is whether implicit or explicit time integration is used. As I ...
Fei Zhu's user avatar
  • 335
1 vote
1 answer
876 views

Stability of forward euler method

I am trying to understand the stability of the forward Euler method. I read there's a model problem to see the stability. $$y'(t) = \lambda y(t) \qquad t \in (0, \infty)$$ $$y(0) = 1$$ then the book ...
BRabbit27's user avatar
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11 votes
4 answers
683 views

Runge-Kutta and Reusing Datapoints

I am trying to implement the fourth order Runge-Kutta method for solving a first order ODE in Python i.e. $\frac{dy}{dx} = f(x,y)$. I understand how the method works, but am trying to write an ...
joshlk's user avatar
  • 273
6 votes
2 answers
446 views

How can I reduce the communication bottleneck of a parallel explicit finite difference scheme?

Suppose i was trying to solve a parabolic PDE (heat equation) on a rectangular domain using an explicit finite difference scheme. I am storing my solution vector in a matrix form (because it closely ...
Paul's user avatar
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17 votes
1 answer
1k views

When should implicit methods be used in the integration of hyperbolic PDEs?

Numerical methods for solving PDEs (or ODEs) fall into two broad categories: explicit and implicit methods. Implicit methods allow larger stable timesteps but require more work per step. For ...
David Ketcheson's user avatar