3.1.1 Introductory example
First of all, let us define some conventions and notations. Assume that the right-hand side fi, in Equation (3), computed on element Ωi, can be split into the volume contribution fi,i and the left and right interface contributions fi − 1,i and fi + 1,i, depicted in Figure 3, that use information from the neighboring elements:
The volume and interface terms for each element at each stage k of an ERK method may be defined as follows:
At each stage of the ERK method, . For a classical singlerate ERK method, because the normals at the interface between two neighboring elements are opposite each other. This property ensures global conservation of the fluxes after each iteration of the method.
Figure 3. one-dimensional unstructured mesh with interface fluxes, fi,i − 1 and fi − 1,i, between neighboring elements Ωi − 1 and Ωi. is applied on element Ω0, whereas is used for elements Ω1, Ω2, and Ω3.
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Consider the Second-order accurate SSP ERK base method RK2a represented in Table 2 (a) and the mesh depicted in Figure 2. The key idea is to extend the RK2a method, , to a four-stage adapted Butcher tableau , Table 2 (b), where the base method is repeated twice on the same time interval. Actually, is strictly equivalent to the base method if it is used on all elements. The Butcher tableau shown in Table 2 (c) corresponds to the base method applied twice successively with the same time step . Actually, this Butcher tableau contains implicitly the update for . In other words, RK2a is applied at first time to un to obtain , that corresponds to time , and then again to to compute un + 1. The methods and have now the same number of stages, and therefore, it enables the transition between the two types of elements.
Table 2. Butcher tableaus corresponding to (a) the two-stage, Second-order base method, (b) the buffer-adapted method, and (c) the two-stage, Second-order method applied twice successively with a twice as small time step.
At the first stage of the coupled methods, there are no ambiguities. It is identical to apply the same base method RK2a everywhere. The are all computed at the same intermediate time level . Virtual incoming fluxes, f − 10 and f43, are supplied at the boundary of the domain as represented in Figure 3. It is assumed that the virtual element Ω − 1 (resp. Ω4) is of the same type, same size, and uses the same ERK method, as its neighbor Ω0 (resp. Ω3).
At the second stage, the intermediate time levels are not the same on each element, that is, , whereas for i = 1,2,3.
About the conservation of the fluxes at the interfaces between elements, we can check that for i = 1,2,3 and k = 1,2,3,4. Because the b vectors of the Butcher tableaus are equal, ba = b2b, for all elements, the sum of the fluxes cancels at each interface:
The so-called first-order and Second-order conditions are verified for the two methods considered separately . At the critical interface between Ω0 and Ω1, the order of the coupling between and has to be considered. The first-order coupling conditions are implicitly satisfied. It can be verified that the Second-order PRK coupling conditions, that is,
are satisfied . The RK2a multirate method of Constantinescu is thus globally Second-order accurate. Indeed, for PRK methods, the global order is defined as the minimum among the orders of the two methods considered separately and the order of their coupling .
The strategy of Constantinescu  may be used to manage different integer time step ratios. A time step ratio κ = 2 between the different multirate groups seems to be sufficient for our target applications. Stable time steps of two neighboring cells are assumed to be relatively close for the vast majority of the mesh elements. This multirate approach may be extended, not only to any s-stage ERK base method, as shown in Table 3 , but also to multiple levels of refinement. It is nevertheless required, for an s-stage base method, that a buffer region of at least s connected elements separates two bulk groups. It is only at that distance that it is possible to collapse the adapted method into the base method. This general property can be proved using the same arguments as in the earlier introductory example. Imbricated multirate groups for buffers of size 2,3, and 4 are illustrated around the Holbourne island in Figure 13(a)–(c).
Elements are connected through their interfaces (nodes in 1D, segments in 2D, and faces in three-dimensional (3D)) in a DG formulation. This is a major advantage, in the context of multirate methods, compared with the standard continuous FEM where all types of elements are connected through nodes. Accordingly, buffer regions are generally considerably larger than in the discontinuous case, and more elements need twice as many operations as required by their stable time step. The efficiency of the multirate methods is therefore lower. Another issue, with continuous elements, is the handling of the mass matrix that is not block diagonal and thus couples the whole solution. This would complicate the use of several time steps on different multirate groups. However, we did not investigate in practice the multirate approach for continuous finite elements.
Table 3. Butcher tableaus for (a) the arbitrary s-stage explicit Runge–Kutta base method, (b) the adapted buffer method, and (c) the base method with half of the time step applied twice successively.
Because this multirate strategy is based on the PRK method, the order of the coupled method can be obtained as the minimum among the base methods used and the order of their coupling [30, 31]. Constantinescu  has shown that the MPRK-2 schemes, defined by the Butcher tableaus in Table 3, are (1) Second-order accurate if the base method is at least Second-order accurate and (2) have at most a Second-order accurate coupling regardless of the order of the base method. The third-order coupling conditions are never all satisfied for this multirate strategy . It is actually at each critical interface, between a buffer group and a more constrained bulk group, that the coupling reduces to Second-order accuracy.
In spite of the order restrictions, the MPRK-2 schemes present the advantage of being conservative. It is shown in  that any PRK method with the same weights (ba = b2b) is conservative. In particular, MPRK-2 (described by Table 3) is conservative.