# 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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### 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 ...
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### 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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### 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 ...
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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 ...
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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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### 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 ...
1 vote
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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 $f$ ...
1 vote
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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 ...
1 vote
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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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1 vote
### 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 ... 