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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Implicit-Explicit Operator Splitting Scheme
I am trying to solve the 2D advection-diffusion equation in cylindrical coordinates:
$$
\frac{\partial c}{\partial t} = D\left(\frac{\partial^2 c}{\partial r^2} + \frac{1}{r}\frac{\partial c}{\partial ...
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How to implement boundary conditions for the Thomas algorithm
For my variable $U(t,x)$, I have implemented the thomas algorithm with $U_j^i$:
$$ 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?
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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 ...
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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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