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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168 votes
8 answers
136k views

Recommendations for a usable, fast C++ matrix library?

Does anyone have recommendations on a usable, fast C++ matrix library? What I mean by usable is the following: Matrix objects have an intuitive interface (ex.: I can use rows and columns while ...
23 votes
1 answer
14k views

Why is Newton's method not converging?

I am using PETSc's nonlinear solver package SNES to solve a system of nonlinear equations obtained by discretizing a partial differential equation. How can I determine why the solver is not converging ...
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17 votes
1 answer
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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 ...
16 votes
2 answers
327 views

What are the best practices for algorithms and implementation of multi-physics simulations?

Multi-physics simulation involves coupling multiple "physics", often with different space and/or time scales. Additionally, the single-physics codes are often written by different teams. The most ...
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15 votes
2 answers
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Implicit finite difference schemes for advection equation

There are numerous FD schemes for the advection equation $\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}=0$ discuss in the web. For instance here: http://farside.ph.utexas.edu/teaching/...
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15 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 ...
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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 ...
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10 votes
1 answer
832 views

cuda and numerical methods with implicit time discretization

I am looking to port some code that resolves a set of partial differential equations (PDE) by the finite volume method in IMPLICIT form (for the time discretization). As result there is a tridiagonal ...
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9 votes
1 answer
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Matlab solution for implicit finite difference heat equation with kinetic reactions

I am trying to model heat conduction within a wood cylinder using implicit finite difference methods. The general heat equation that I'm using for cylindrical and spherical shapes is: Where p is the ...
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8 votes
1 answer
917 views

Linearized implicit time stepping

Consider the general FD implicit time stepping scheme $\frac{x_{t+1} - x_t}{\Delta t} = f(x_{t+1})$, where $x$ is the vector variable of interest and $f$ is some function, generally non-linear. ...
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7 votes
2 answers
491 views

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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7 votes
2 answers
345 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 ...
6 votes
1 answer
836 views

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}...
6 votes
1 answer
173 views

Can heat distribution in an optical element irradiated by laser be oscillating?

I am modelling a heat distribution in optical element irradiated by laser. System is radially symmetric, and element is thin, i.e. heat value depends only on distance from center. Heat is received via ...
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5 votes
2 answers
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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 ...
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5 votes
2 answers
10k views

How to implement Newton's method for solving the algebraic equations in the backward Euler method

Can you explain me how does the backward Euler method works? I have seen the formula and try to understand the method, but what I can't understand is why and how to use the Newton-Rapson method. Do ...
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5 votes
1 answer
363 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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5 votes
1 answer
141 views

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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5 votes
1 answer
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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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5 votes
1 answer
518 views

Does ADI/Split-operator change the stability properties of the Crank-Nicholson method?

I'm using the Crank-Nicholson method to solve the time-dependent Schrödinger equation with the split-operator method. I'm getting some weird results that are probably the result of a bug somewhere in ...
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4 votes
1 answer
281 views

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 ...
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4 votes
2 answers
265 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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4 votes
1 answer
281 views

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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0 answers
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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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4 votes
0 answers
718 views

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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1 answer
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How to obtain an implicit finite difference scheme for the wave equation?

Suppose I had the following problem: $U_{tt}=U_{xx}+U_{yy}$ in $\Omega=[0,1]\times[0,1]$ $U(x,y,0)=f(x,y)$ $U_{t}(x,y,0)=g(x,y)$ $U=0$ on $\partial \Omega$ I know that there is an explicit ...
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3 votes
1 answer
755 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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3 votes
1 answer
269 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 ...
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3 votes
1 answer
1k views

Implicit heat diffusion with kinetic reactions

I am using the implicit finite difference method to discretize the 1-D transient heat diffusion equation for solid spherical and cylindrical shapes: $$ \frac{1}{\alpha}\frac{\partial T}{\partial t} = ...
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3 votes
1 answer
253 views

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 ...
2 votes
1 answer
671 views

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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2 votes
1 answer
2k views

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\,(...
2 votes
1 answer
368 views

CFL evolution techniques for Implicit methods

I am working on implicit schemes for Euler equations. Implicit methods allow one to use large CFL values, but is there some way to evolve CFL number from a much smaller value than desired value to ...
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2 votes
1 answer
258 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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2 votes
1 answer
76 views

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 ...
2 votes
2 answers
115 views

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. ...
2 votes
0 answers
61 views

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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2 votes
0 answers
58 views

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,...
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1 vote
1 answer
111 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 ...
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1 vote
1 answer
68 views

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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1 vote
1 answer
141 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 ...
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1 vote
1 answer
43 views

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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1 vote
1 answer
246 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 ...
1 vote
1 answer
233 views

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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1 vote
1 answer
379 views

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}...
1 vote
1 answer
551 views

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 ...
user avatar
1 vote
1 answer
269 views

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) ...
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1 vote
2 answers
8k views

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 ...
1 vote
1 answer
156 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.,} \...
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1 vote
0 answers
32 views

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 ...