I want to check order of accuracy of MUSCL-Hancock scheme in time on a 1D problem with exact smooth solution. The time-marching is dealt with using the Strang splitting. We can confirm that this method is second-order accurate in space by calculating L_1 and L_infty norms. But I do not how to check accuracy in time. who can give me some advice. Thanks.

Dear Legend,

The idea of obtaining the order of accuracy is essentially similar either in space or time. All you need to do is to take a reasonably good grid (say 200 points or so), and run the numerical simulation for a sufficiently small time step, say t_s. Now, keep the grid fixed and run the problem on the mesh for a sequence of time steps t1,t2,.....,tn, such that t1=2t2=4t3=...=2^n t(n+1). This is a sequence of progressively smaller time steps and the code needs to be run upto the same final time T. The solution at time t_s is closest to the exact solution, and all other solutions are numerical solutions and the error can be found out on the given grid. Now by computing the suitable error norms, the temporal accuracy is easily determined. Clearly, t_s is smaller than t^(n+1) which is the smallest time step for which the numerical simulation is performed. Usually t_s is chosen a 1/10th of t^(n+1). For more clarifications just google "Temporal accuracy of numerical scheme" and you shall get the details.

Hope this helps.

Regards,

Ganesh