## Generalised Galerkin methods for hyperbolic equations.(English)Zbl 0551.65076

The mathematics of finite elements and applications IV, MAFELAP 1981, Proc. Conf., Uxbridge/Middlesex 1981, 421-430 (1982).
[For the entire collection see Zbl 0496.00017.]
Time-dependent problems $$\partial_ tu=Lu$$ are solved approximatively by first using a time-differencing procedure (e.g. a linear k-step method with parameters $$\{\alpha_{\nu},\beta_{\nu};\nu =0,...,k\})$$ and then by designing a finite element method by the expansion $$U^ n(x)=\sum_{j}V^ n_ j\Phi_ j(x)$$ with basis functions $$\Phi_ j(x)$$. In one dimension $$(Lu=\partial_ xf(u))$$ and in two dimensions $$(Lu=a\partial_ xu+b\partial_ yu)$$ different so-called Petrov- Galerkin methods of the form, $$(<.,.>$$ denotes the usual $$L_ 2$$ inner product over the space domain), $<\sum^{k}_{\nu =0}\{\alpha_{\nu}U^{n+\nu}-\Delta t\quad \beta_{\nu}L(U^{n+\nu})\},\quad \psi_ i>=0\quad \forall i$ are developed applying special test functions $$\psi_ i$$. Moreover, generalized Galerkin methods using characteristics are considered.
Reviewer: H.Antes

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35L50 Initial-boundary value problems for first-order hyperbolic systems

Zbl 0496.00017