timeis the visualization on the screenbut for that purpose one can visualize only subset of the time steps howeverthere are occasions when you need to take larger time steps with the diffusion equationespecially if interest is in the longterm behavior as you must then turn to implicit methods for odes these methods require the solutions of linear systemsif the underlying pde is linearand systems of nonlinear algebraic equations if the underlying pde is non-linear the simplest implicit method is the backward euler schemewhich puts no restrictions on for stabilitybut obviouslya large leads to inaccurate results the backward euler scheme for scalar ode utreads unc un unc tnc this equation is to be solved for unc if is linear in uit is linear equationbut if is nonlinear in uone needs approximate methods for nonlinear equations (chap in our casewe have system of linear odes ( )-( the backward euler scheme applied to each equation leads to un unc tnc / uni unc unc unc unc gi tnc / unc unn nc unc gi tnc ( ( ( which this is system of linear equations in the unknowns unc is easy to realize by writing out the equations for the case collecting all the unknown terms on the left-hand side and all the known terms on the right-hand sideunc un tnc / unc unc unc tnc / nc unc unc tnc unc ( ( ( system of linear equations like thisis usually written on matrix form au bnc where is coefficient matrixu unc nn is the vector of unknownsand is vector of known values the coefficient matrix for the case ( )-( becomes