Abstracts A comparison of triple jump and Suzuki fractals for obtaining high order from an almost symmetric Strang splitting scheme

A comparison of triple jump and Suzuki fractals for obtaining high order from an almost symmetric Strang splitting scheme Lukas Einkemmer, Alexander Ostermann We consider the time discretization of ordinary and partial differential equations. More specifically, we assume that the considered problem can be written as the following abstract Cauchy problem (1) u′ = Au+B(u), u(0) = u0. In this context splitting methods can be applied if the partial flows generated by A and B have an analytical representation, or if an efficient algorithm for finding their exact solution is known. However, the assumption that the partial flow generated by the nonlinear operator B can be computed exactly is usually a very strong requirement. To remedy this deficiency of classic splitting methods, we propose and analyze splitting schemes which approximate the partial flow generated by B by that of an inhomogeneous linear differential equation. That is, we consider the linearized problem given by (2) v′ = b(u?)v + d, where we assume that, once a value u? is substituted, the flow corresponding to (2) can be computed efficiently. In this context, we can formulate the (classic) Strang splitting scheme as follows (3) Mτ (u0) = e τ 2φ b(u1/2) τ (e τ 2u0), u1/2 = φ b(u0) τ 2 (e τ 2u0), where the flow corresponding to equation (2) is denoted by φ b(u?) t ; τ is the step size. The Strang splitting above is no longer symmetric. The lacking symmetry of the method does not severely affect performance; however, if composition is used as a means to construct higher order methods, symmetry is a desirable property. To remedy the lack of symmetry we consider the symmetric scheme u1 = Sτ (u0) given by the solution of the following implicit equation u1 = e τ 2A ◦ φ1 τ 2 (u1/2). Note that solving this equation is computationally not attractive in practice. Therefore, we proposed to employ fixed-point iterations to compute an approximation to u1 (see [1] and [2]). The resulting one-step methods are not symmetric but they are symmetric up to a given order (a precise definition is given in [1]). Thus, it is possible to construct composition methods of arbitrary (even) order. In [1] and [2] we have performed numerical simulations for a variety of ordinary and partial differential equations. To obtain methods of order four we have used the well-known triple jump composition. However, at least for some ODEs, the so called Suzuki fractals give better performance (see, e.g., [3]). In this case, five