Implicit Integration Methods

Implicit integration methods are used with systems that are reversible in time. The general form for implicit integration is
$$Y_{new}=Y_0+hf(Y_{new})$$
which can also be re-written as:
$$x(n+1)-x(n)-h*Dx(n+1)=0$$
This can be thought of as taking a backward step from the future timestep value that will allow you to arrive at the current timestep. The stability region for the implicit Euler method is the entire left half of the complex plane, excluding the imaginary axis. This means that the implicit Euler method is unconditionally stable, and will always produce a solution that remains bounded for any value of the step size, h. Implicit solvers are more stable but require more computations per step. Implicit methods are also better at solving stiff problems. Implicit solvers are stable for any step size.[^3] For each step, sets of nonlinear equations must be solved, therefore it may take a lot of computational time[^3] We can make an explicit method into an implicit method by taking explicit terms and moving them to the opposite side of the equation, for example the explicit function in reactor thermal models:
$$T^{n+1}=T^n+\frac{\Delta tk(T^n)}{\rho c_p}\nabla^2T^n+\frac{\Delta tq^{”’,n}}{\rho c_p}$$
Can be made implicit:
$$T^{n+1}-\frac{\Delta tk(T^{n+1})}{\rho c_p}\nabla^2T^{n+1}+\frac{\Delta tq^{”’,n}}{\rho c_p}$$
Then the matrix form can be solved. Implicit integration methods reduce their dependence on timestep sizes, at the cost of greater numerical diffusion in the solution.

[[Backward Euler]]- implicit integration method
[[Matlab ode1be]]
[[Matlab ode14bx]]
[[SPICE]]- Spice need to use implicit integration methods for transient analysis.
[[Implicit vs Explicit Integration Examples]]
Stiffness Ratio – implicit integration is better at solving these
CFD Grid Fin Simulation – can use a implicit solver for CFD
[[Advanced Continuous Simulation Language]] – implicit state derivatives are not recommended.
RELAP5 – uses semi-implicit methods
[[Explicit Integration Methods]] -the implicit version of this example is shown above
[[Fully-Implicit Transient Newton Krylov Methods]]
[[Delta Guidance]] – termed implicit