Questions tagged [explicit-methods]

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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 ...
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1answer
45 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 ...
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1answer
70 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 ...
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1answer
155 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 ...
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1answer
95 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 ...
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1answer
83 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 ...
5
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1answer
255 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 ...
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1answer
52 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 ...
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1answer
557 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 ...
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1answer
163 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?
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0answers
124 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 ...
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0answers
219 views

Limit to precision of step-size

When solving an equation of the following form: $$ \begin{aligned} \frac {\partial A}{\partial t} &= EB - A \\ \frac {\partial B}{\partial t} &= EA - B \\ \frac {\partial E}{\partial t} &...
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0answers
282 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 ...
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1answer
368 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 ...
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0answers
40 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....
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2answers
85 views

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 ...
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2answers
2k 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 ...
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3answers
7k 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 ...
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1answer
436 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 ...
11
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4answers
477 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 ...
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2answers
404 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 ...
16
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1answer
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 ...