Towards an Effective Algorithm. Practical Issues

Abstract

In this chapter, we present some practical issues which are useful for the implementation of our adaptive strategies. Efficient coding of Rosenbrock methods and dynamic multilevel techniques in two and three dimensions are addressed. We consider a generalized class of partial differential algebraic equations (PDAEs) of the linearly implicit form

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGibWaaeWaa8aabaWdbiaadshacaGGSaGaamyDaaGaayjkaiaa % wMcaaiabgkGi2+aadaWgaaWcbaWdbiaadshaa8aabeaak8qacaWG1b % Gaeyypa0JaamOramaabmaapaqaa8qacaWG0bGaaiilaiaadwhaaiaa % wIcacaGLPaaacaGGSaGaamyDamaabmaapaqaa8qacaaIWaaacaGLOa % GaayzkaaGaeyypa0JaamyDa8aadaWgaaWcbaWdbiaaicdaa8aabeaa % k8qacaGGSaGaaGimaiabgYda8iaadshadaadaaqaaiabgYda8aaaca % WGubGaaiilaaaa!5253!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$H\left( {t,u} \right){\partial _t}u = F\left( {t,u} \right),u\left( 0 \right) = {u_0},0 < t\underline < T,$$

(V.1)

which is met often in practical computations. As for differential-algebraic equations (DAEs), a classification based on an index plays a key role in the description of PDAEs. This classification is an important topic in current research and there are many open problems even in the linear case. For instance, different types of spatial semi-discretizations may lead to finite dimensional DAEs of different indices. Moreover, such systems may have an index jump [42, 109].