A simplified Cauchy-Kowalewskaya procedure for the local implicit solution of generalized Riemann problems of hyperbolic balance laws

A universal centred high-order method based on implicit Taylor series expansion with fast second order evolution of spatial derivatives

“…For evaluating the time derivative, we use the Cauchy-Kowalewskaya procedure. We can use the recently simplified version of the Cauchy-Kowalewskaya procedure in [27]. However, once the Cauchy-Kowalewskaya functionals are available the resulting scheme becomes a very efficient one.…”

“…Notice that (6) requires not only the predictor but also the derivative of this. Since the value Q(ξ, τ ) at each quadrature point in space and time is known, we use an interpolation polynomial of Q as in [27], that is, at each quadrature point τ j in time we use the set {Q(ξ l , τ j )} l to carry out a Lagrange interpolation. Then, we use the derivative of this polynomial to approximate ∂ x Q.…”

“…However, Toro and Montecinos [34] have shown how implicit Taylor series can be adapted for dealing with stiff source terms. Recently, in [27] a simplified Cauchy-Kowalewsky which only requires the Jacobian matrices and a strategy for evaluation their derivatives are enough to generate all the functional in a very efficient way.…”

“…Despite a new strategy for approximating the Cauchy-Kowalewskaya functional is available in [27], the existence of the closed forms given by the conventional Cauchy-Kowalewskaya functional should be as efficient as the simplified version. For this reason, in this work a practical strategy for implementing the conventional procedure for general hyperbolic balance laws, based on the syntaxes of the well known, GNU computer algebra system Maxima, [28], is reported and so this is the approach implemented in this paper.…”