Questions tagged [implicit-methods]

Implicit methods are timestepping methods that use an inversion at every timestep. This allows for much better stability properties than explicit methods, though it comes with a serious speed penalty in some cases. Examples of implicit methods include Backward Euler and Crank-Nicholson.

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How to implement boundary conditions for the Thomas algorithm

For my variable $U(t,x)$, I have implemented the thomas algorithm with $U_j^i$: $$ a(x)U_{j-1}^{i+1}+ b(x)U_j^{i+1} + c(x)U_{j+1}^{i+1} = d(x)U_j^{i} $$ Then $\textbf{A}$ is a tridiagonal vector with ...
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Unable to solve numerically this system of differential equation

I'm trying to obtain the graph of x(y) from the following system : Therefore I tried to solve this system using an Euler Method : ...
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2nd-order backward difference approximation for temporal terms of Euler–Bernoulli beam

I am trying to solve the Euler–Bernoulli beam numerically in structural dynamics analysis: $$\frac{\partial^2w(x,t)}{\partial t^2}= \frac {q(x,t)}{\mu(x)}-\frac {1}{\mu(x)}\frac {\partial^2}{\partial ...
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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. ...
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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
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Does this second-order implicit Runge-Kutta method have a name?

I am studying the time-integration of the following paper, Young, L. C. (1981). A finite-element method for reservoir simulation. Society of Petroleum Engineers Journal, 21(01), 115-128. A copy (PDF)...
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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 $f$ ...
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Accuracy of the Crank-Nicolson method for non-linear, inhomogeneous heat equation

I am currently coding a solution to the following PDE: $\frac{\partial T }{\partial t} =\frac{\partial}{\partial \theta}(A(\theta ,\phi )\frac{\partial T }{\partial \theta}) +\frac{\partial }{\partial ...
mathbruh67's user avatar
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How can I derive a second order implicit method for this coupled ODE update?

I appreciate this might be an easy question, but I've managed to get myself quite thoroughly confused I'm trying to solve a system of physics equations that look as follows $$ \frac{\partial \mathbf{E}...
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Are stiffness and instability equivalent?

To the best of my knowledge, stiffness of ordinary differential equations is difficult to capture but can be roughly described as problems where explicit methods don't work while implicit ones do. ...
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Calculating the species mass consumption from implicit reaction-term in diffusion-reaction equation

The 1D diffusion equation with a chemical source term has the following form: $$\frac{\partial Y}{\partial t} = D \frac{\partial^2 Y}{\partial x^2} - k Y,$$ where $Y$ is the molar concentration of the ...
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Stiff ODE solver in the web browser

I'd like to make a web application that lets people play with solving ODE systems, changing parameters with sliders etc. but instead of doing the computations on the server side, solving the equations ...
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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 ...
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Implicit methods for variable coefficients based on equations of state

For example I have an equation that goes something like $ \partial_t \rho = -\nabla\cdot (\rho u) + \nabla \cdot(D(\rho, T) \nabla \rho) + \rho_s $ ($\rho, \rho_s, u, T$ are coupled with a few other ...
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How to fill matrix entries for two-dimensional implicit finite-difference for the general case

If I have derived a finite-difference formula for a 2D problem, for example something like: $af_{i,j}+bf_{i-1,j}+cf_{i,j-1}+df_{i-1,j-1}=g_{i,j}$ where f is the unknown function on a grid and ...
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Understanding butcher tableau when it comes to implicit methods

I've been learning about butcher tables and am having some difficulty understanding how to translate them when it comes to implicit methods. Specifically, I'm looking at backwards Euler: \begin{array}...
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Can Taylor methods be used effectively on stiff ODEs?

Cleve Moler has stated that "all numerical methods for stiff odes are implicit." However, I don't know whether this statement is a mathematical fact, or an simply an observation. Moreover, many ...
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GMRES vs Newton-GMRES for Solving nonlinear PDE's

Often when numerically solving nonlinear PDE's using method of lines approach with an implicit integrator a system of nonlinear equations have to be solved. To be more specific, let's say we have ...
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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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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
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Mass matrix and BDF time integration

I have a system of nonlinear equations on the general form: \begin{align} \mathbf{M}(\bar{y})\dot{\bar{y}} =\bar{f}(\bar{y},t) \end{align} Where $\mathbf{M}(\bar{y})$ is a matrix and $\bar{f}$ is a ...
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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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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 ...
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Solving a PDE implicitly by iteration in python

Connected to this question here on Computational Science, I've posted a follow-up question on how to solve a PDE using an implicit scheme like Crank-Nicholson in general in this question on SO. But I ...
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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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Implicit solution to Sylvester equation

Suppose a matrix $M\in\mathbb{R}^{n\times n}$ is defined as the solution to a Sylvester equation $$AM+MB=C,$$ for some fixed (known) matrices $A,B,C$. In the regime where $n$ is large, we may with ...
Justin Solomon's user avatar
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Adding Non-Linear source term to 2d Implicit MATLAB code

I'm running out of time for this code so any help would be greatly appreciated. I am currently coding the 2D heat/diffusion equation in matlab but i'm having trouble adding in the source term. my ...
DesperateStudent's user avatar
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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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Implicit integration for FLIP?

I have problem with volume loss in FLIP simulation. Unfortunately it's necessary to obey the CFL condition when using explicit integration methods (RK2 in my case) to advance particle positions using ...
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Slight Modification to Backward Euler Stiff ODE Solver

I am trying to implement a stiff ODE solver that uses step doubling with the backward Euler method but I want to make a modification and I am unsure how to incorporate it. I am trying to solve $y'=f(...
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How to support or contradict a hypothesis on unconditional stability using numerical optimization

The main motivation behind my next question is that I think I derived a higher order numerical scheme for linear advection equation that is unconditionally stable using Von Neumann stability analysis. ...
Peter Frolkovič's user avatar
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2 answers
286 views

Textbook/Manual on Implicit FEM Methods

I've recently been interested in learning about implicit finite element methods. I've found this post, but I'd like to learn more about them, specifically about how simulations like these from LS-DYNA ...
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Solve an ODE with positivity-preserving property unconditionally

I have an ODE for a scalar function $u=u(t)$ of the form: $$ \frac{du}{dt}=L(u). $$ Here the function $L=L(u)$ satisfies: $$ L(0)=0, \quad L'(u)\le0. $$ Then it is easy to see that the solution $u=u(t)...
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How to solve bring my implicit equation to closed form? [duplicate]

I have a simulated system as shown in the following image: $L_0$ is attached to two other bodies $L_1$ and $L_2$. Furthermore, body $L_3$ is also in the simulation (it is attached to $L_2$ but that ...
WhiteTiger's user avatar
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Why is this method for simulating a system of springs and masses unstable?

I have a computer simulation system of bodies connected by springs, so their movement is governed by: $x_{n+1} = x_n-\Delta tk(x_n-r)$ Where $r$ is the idea distance between every two bodies, and $\...
WhiteTiger's user avatar
2 votes
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Resolving a stiff hyperbolic problem with Neumann boundary conditions

I am trying to numerically resolve the equation for an Euler-Bernoulli beam that is inextensible, unshearable, and subject to planar deformations: $$\rho I(s) \frac{\partial^2 \theta}{\partial t^2}(s,...
namu's user avatar
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1 answer
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LU-SGS and boundary conditions

I am trying to understand how boundary conditions are implemented when one uses the nonlinear LU-SGS algorithm for Euler equations. Most papers describe the Gauss-Seidel sweep over mesh cells, but do ...
Martin Vymazal's user avatar
5 votes
2 answers
2k views

Why is Crank-Nicolson considered implicit in time?

From Wikipedia: Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation ...
Sparkler's user avatar
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Implicit ODE solver with discontinuous derivatives

I want to implement an implicit ODE solver, but don't know what to do when the differential equations (DEs) have discontinuities of the form: More common type: $$\lim_{x\rightarrow 0}\frac{\sin(x)}{x}...
Piotr Sokol's user avatar
2 votes
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281 views

Not getting correct numerical solution for Advection-Diffusion-Reaction eqn

Objective: I am trying to numerically solve $C(x,y,t)$ from the following advection-diffusion-reaction equation in 2D space (x,y) and time. I will be testing my numerical solution with an approximate ...
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ADR equation implicit solution: Penta-diagonal matrix for a 2D $N\times N$ system

Objective: I am trying to simulate the following advection-diffusion-reaction equation in 2D space (x,y) and time. $$\begin{align} \text{ADR Equation: }\frac{\partial C}{\partial t} + \nabla\left(v.C ...
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Implicit time integrator for Chebyshev collocation method for linear hyperbolic system

I want to solve linear hyperbolic system using Chebyshev collocation method. As this method puts severe constraint on the time step for the explicit time integration, I decided to switch to implicit ...
Dmitry Kabanov's user avatar
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Spectral Coefficients of Implicit Finite Difference Solution

This is something I've been trying to figure out for a long time, and all I have is vague numerical results. I'm trying to answer the following question analytically: Suppose I have a time dependent ...
Amir Sagiv's user avatar
7 votes
2 answers
398 views

Initializing implicit linear multistep methods

A sixth order backward differentiation formula (BDF) need six (five plus initial value) previous solutions to get started. How I can get these previous solutions? I need a method accurate to sixth ...
user3368561's user avatar
1 vote
1 answer
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CFD: Doubt with time convergence in advection fully implicit upwind scheme

I'm trying to solve an advection - convection problem using an implicit upwind scheme - you can see here the finite difference discretization used. I start the model (built from scratch on Scilab) ...
cfder's user avatar
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Fluent time step size (transient implicit solver)

I am studying fluid-structure interaction across integrated circuits packaging by using Ansys Fluent (I'm a beginner). Currently I had an issue about the time step size. When I set time step size of ...
user11679's user avatar
4 votes
1 answer
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Non-conservative implementation implicit Euler

In Matlab R2013a I have implemented the Implicit Euler (time) integration scheme. To find the $x^{n+1}$ value I use fixed point iterations: $x^{n+1} = \Delta t f(x^{n+1}) + x^n$ To test this, I use ...
Rhino's user avatar
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How can I implement the implicit Euler method for a small nonlinear system of ODEs?

I am trying to solve a system of coupled ODEs: $$ \begin{align} \frac{dn_A}{dt} & = e\left[j(t) - f\, θ_H\sinh\left(\frac{g\,n_A}{T}\right)\right] \\ \frac{dθ_H}{dt} & = a\left[bP\,(...
Ushnik Mukherjee's user avatar
1 vote
1 answer
173 views

Implicit finite difference scheme for LWR-v PDE

I want to discretize the LWR(Lighthill-Whitham-Richards)-v partial differential equation (as shown in 1) $$ \frac{\partial u}{\partial t} + \frac{\partial R(u)}{\partial x} = 0 \quad \text{i.e.,} \...
sid81's user avatar
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How to add reaction and source terms to a diffusion PDE solver written with MATLAB's pdepe function?

I have the following system of equations which I'm trying to solve using Matlab's pdepe solver. The 1-D spherical heat diffusion equation with heat generation (...
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