An Approximate Lax–Wendroff-Type Procedure for High Order Accurate Schemes for Hyperbolic Conservation Laws

On approximate implicit Taylor methods for ordinary differential equations

“…is understood as acting on each component of f ( P k n (h)) separately.) There exist constants β k,R j , so that for some integers γ k,R , we can write (see Zorío et al 2017…”

“…Miletics and Molnárka (2004) propose an alternative based on a numerical approximation of the derivatives of f in ODEs of the form u = f (u) for the explicit Taylor method up to fourth order and in Miletics and Molnárka (2005) for the implicit version up to fifth order. Later on, in Baeza et al (2017), a procedure to obtain a numerical approximation of f (u) = f • u was presented to generate arbitrarily high-order Taylor schemes, inspired by an approximate Cauchy-Kovalevskaya procedure developed for systems of conservation laws by Zorío et al (2017), which simplifies the exact version presented by Qiu and Shu (2003). The method presented by Baeza et al (2017) relies on the approximate computation of the terms that appear in the Taylor polynomials, in terms of function evaluations only, avoiding the explicit computation of the derivatives, leading to a method which is simple to implement and outperforms its exact counterpart for complex systems.…”

Compact Approximate Taylor Methods for Systems of Conservation Laws

“…where {x i } are the nodes of a uniform mesh of step ∆x; u n i is a pointwise approximation of the solution at x i at the time n∆t, where ∆t is the time step; andũ (k) i is an approximation of ∂ k t u(x i , n∆t). The strategy followed in [2] to avoid the CK procedure is based on the equalities ∂ k t u = −∂ x ∂ k−1 t f (u). (2.3) that can be easily derived from the equation, if the solutions are assumed to be smooth enough.…”

“…a contact discontinuity with discontinuous tangential velocity J. The sub-indexes (l, r) ∈ {(2, 1), (3,2), (3,4), (4, 1)} indicate the involved quadrants. For shocks and rarefactions an over-arrow indicate the direction (backward or forward).…”

Implicit Multiderivative Collocation Solvers for Linear Partial Differential Equations with Discontinuous Galerkin Spatial Discretizations

“…This is attributed to the original work of Lax and Wendroff from 1960 where they construct a second-order solver by incorporating the second derivative of the PDE into their method [33]. More recently, higher order (i.e., solvers with order greater than two) versions of these solvers have been investigated for finite volume [26], finite difference [41,32,46,12,14,52], and discontinuous Galerkin discretizations [40,22,36]. A large community centered around Arbitrary DERivative (ADER) discretizations has been very successful with constructing arbitrary order explicit solvers for hyperbolic problems in this category [48,51,15,7], and much of their work relies on symbolic software to generate their code base.…”