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

Morton, K. W.; Stokes, A.

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

Keywords:

Time-dependent; time-differencing procedure; linear k-step method; finite element method; Petrov-Galerkin methods

Citations:

Zbl 0496.00017