# 1.7 Stiffness and Implicit Methods

## 1.7.3 Implicit Methods for Linear Systems of ODE's

While implicit methods can allow significantly larger timesteps, they do involve more work than explicit methods. Consider the forward method applied to $$u_ t = Au$$ where $$A$$ is a $$d \times d$$ matrix.

 $v^{n+1} = v^ n + {\Delta t}A v^ n.$ (1.122)

In this explicit algorithm, the largest computational cost is the matrix vector multiply, $$A v^ n$$ which is an $$O(d^2)$$ operation. Now, for backward Euler,

 $v^{n+1} = v^ n + {\Delta t}A v^{n+1}.$ (1.123)

Re-arranging to solve for $$v^{n+1}$$ gives:

 $$\displaystyle v^{n+1}$$ $$\displaystyle =$$ $$\displaystyle v^ n + {\Delta t}A v^{n+1},$$ (1.124) $$\displaystyle v^{n+1} - {\Delta t}A v^{n+1}$$ $$\displaystyle =$$ $$\displaystyle v^ n,$$ (1.125) $$\displaystyle \left(I - {\Delta t}A\right)v^{n+1}$$ $$\displaystyle =$$ $$\displaystyle v^ n,$$ (1.126)

Thus, to find $$v^{n+1}$$ requires the solution of a $$d \times d$$ system of equations which is an $$O(d^3)$$ cost. As a result, for large systems, the cost of the $$O(d^3)$$ linear solution may begin to outweigh the benefits of the larger timesteps that are possible when using implicit methods.