solving partial differential equations bthe backward eulerforward eulerand crank-nicolson methods can be given unified implementation for linear ode au this formulation is known as the ruleunc un /aun aunc for we recover the forward euler methodd gives the backward euler schemeand = corresponds to the crank-nicolson method the approximation error in the rule is proportional to texcept for = where it is proportional to for = the method is stable for all apply the rule to the ode system for one-dimensional diffusion equation identify the linear system to be solved cimplement the rule with aid of the odespy package the relevant object name is thetarulesolver odespy thetarule(rhsf_is_linear=truejac=ktheta= dconsider the physical application from sect run this case with the rule and = for the following values of report what you see filenamerod_thetarule py remarks despite the fact that the crank-nicolson methodor the rule with = is theoretically more accurate than the backward euler and forward euler schemesit may exhibit non-physical oscillations as in the present example if the solution is very steep the oscillations are damped in timeand decreases with decreasing to avoid oscillations one must have at maximum twice the stability limit of the forward euler method this is one reason why the backward euler method (or -step backward schemesee exercise are popular for diffusion equations with abrupt initial conditions exercise compute the diffusion of gaussian peak solve the following diffusion problem@ @ @ @ exp td @ td @ / ( oe ( ( ( the initial condition is the famous and widely used gaussian function with standard deviation (or "width"which is here taken to be smalld : such that the initial condition is peak this peak will then diffuse and become lower and wider compute xtuntil becomes approximately constant over the domain filenamegaussian_diffusion py