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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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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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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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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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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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 ...
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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