# Different values on LHS and RHS using Backward Euler and Crank-Nicolson

I was doing some simulations Fortran to solve two transport equations for soot. One for Mass Fraction and the second for Particle Number Density. The first has an order of magnitude of 10^(-5) and the second has 10^20. I am solving them with Backward Euler and Crank-Nicolson methods for time integration. Also, I'm using a Finite Difference Scheme.

I was having problems with the relative difference of Number Density between time steps. The equation of Number Density has the form: Temporal + Convective = Diffusive + Source.

At the first time step, almost all Left Hand Side (LHS) is temporal term and Right Hand Side (RHS) is dominated by the source term. After this, the temporal term increases so much that the other terms become negligible. So, LHS is much bigger than RHS values.

I am having this problem with both methods. Why does that happen?

• Would it be possible to rescale one or more variables so that there is a smaller difference between the order of magnitude of each variable? See dealii.org/8.4.0/doxygen/deal.II/step_32.html for an example (where rescaling is done) in the context of finite element simulations of convection in the Earth's mantle. – Kyle Apr 7 '18 at 20:28