Introduction and study of fourth order theta schemes for linear wave equations

A new class of high order, implicit, three time step schemes for semi-discretized wave equations is introduced and studied. These schemes are constructed using the modified equation approach, generalizing the θθ-scheme. Their stability properties are investigated via an energy analysis, which enables us to design super-convergent schemes and also optimal stable schemes in terms of consistency errors. Specific numerical algorithms for the fully discrete problem are tested and discussed, showing the efficiency of our approach compared to second order θθ-schemes.