View source for B-convergence

The theory of B-convergence predicts the convergence properties of discretization methods, implicit Runge–Kutta methods in particular, applied to initial value problems for systems of non-linear ordinary differential equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b1100101.png" /> (cf. [[Runge–Kutta method|Runge–Kutta method]]; [[Initial conditions|Initial conditions]]; [[Differential equation, ordinary|Differential equation, ordinary]]).

==Problem class and historical background.==
Concerning the class of ordinary differential equations considered, the underlying assumption is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b1100102.png" /> satisfies a one-sided Lipschitz condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b1100103.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b1100104.png" /> (the so-called one-sided Lipschitz constant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b1100105.png" />) is of moderate size, the initial value problem is easily proved to be well-conditioned throughout. Of particular interest are stiff problems (cf. also [[Stiff differential system|Stiff differential system]]). Stiffness means that that the underlying ordinary differential equation admits smooth solutions, with moderate derivatives, together with non-smooth ( "transient" ) solutions rapidly converging towards smooth ones. In the stiff case, the conventional Lipschitz constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b1100106.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b1100107.png" /> inevitably becomes large; therefore the classical error bounds for discretization methods (which depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b1100108.png" />) are of no use for an appropriate characterization and analysis of methods which are able to efficiently integrate a stiff problem. Thus there is a need for a special convergence theory applicable in the presence of stiffness.

The idea to use <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b1100109.png" /> (instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001010.png" />) as the problem-characterizing parameter goes back to [[#References|[a7]]], where it was used in the analysis of multi-step methods. The point is that stiffness is often compatible with moderate values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001011.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001012.png" />. In the same spirit, the concept of B-stability was introduced in [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]] in the context of implicit Runge–Kutta methods, and an algebraic criterion on the implicit Runge–Kutta coefficients entailing B-stability was derived ( "algebraic stability in numerical analysisalgebraic stability" ). The notion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001013.png" />-stability enables realistic estimates of the propagation of inevitable perturbations like local discretization errors.

==The concept of B-convergence.==
In the convergence theory of implicit Runge–Kutta methods applied to stiff problems, stability is essential but also the analysis of local errors is non-trivial: straightforward estimates are affected by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001014.png" /> and do not reflect reality. For a simple scalar model class, the local error of implicit Runge–Kutta schemes was studied in [[#References|[a13]]]. It turned out that the order observed for practically relevant stepsizes is usually reduced compared to smooth, non-stiff situations.

In [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], the convergence properties of implicit Runge–Kutta schemes are studied and the notion of B-convergence is introduced. A B-convergence result is nothing but a realistic global error estimate based on the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001015.png" /> but unaffected by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001016.png" />. Besides relying on B-stability, the essential point are sharp local error estimates which require a special internal stability property called BS-stability. The latter can be concluded from a certain algebraic condition on the Runge–Kutta coefficients ( "diagonal stability in numerical analysisdiagonal stability" ). Explicit error bounds have been derived for Gauss, Radau-IA and Radau-IIA schemes; the corresponding  "B-convergence order"  is in accordance with the observations from [[#References|[a13]]]. B-convergence results for Lobatto-IIIC schemes are given in [[#References|[a14]]].

An overview on the  "B-theory"  of implicit Runge–Kutta methods is presented in [[#References|[a8]]]. Another relevant text is [[#References|[a12]]].

==Further developments.==
Concerning the relevance of the B-theory for stiff problems, there remains a gap. The point is that for most stiff problems there is a strong discrepancy between the local and the global condition: Neighbouring solutions may locally strongly diverge, such that the problem is locally ill-conditioned. This is a transient effect, leaving the good global condition unaffected. But it inevitably implies that the one-sided Lipschitz constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001017.png" /> is strongly positive (like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001018.png" />). For details, cf. [[#References|[a1]]], where it is shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001019.png" /> remains moderate only for a restricted class of stiff problems, namely with Jacobians <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001020.png" /> that are  "almost normal" . In general, however, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001021.png" /> is large and positive, and the B-convergence bounds based on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001022.png" /> become unrealistically large.

As a consequence, not even linear stiff problems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001023.png" /> are satisfactorily covered. In [[#References|[a2]]] the B-theory is extended to semi-linear stiff problems of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001025.png" /> has a smoothly varying eigensystem and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110010/b11001026.png" /> is smooth. However, this does not cover a sufficiently large class of non-linear problems. Current work concentrates on a more natural, geometric characterization of stiffness; cf., e.g., [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Auzinger,   R. Frank,   G. Kirlinger,   "A note on convergence concepts for stiff problems"  ''Computing'' , '''44'''  (1990)  pp. 197–208</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Auzinger,   R. Frank,   G. Kirlinger,   "An extension of B-convergence for Runge–Kutta methods"  ''Appl. Numer. Math.'' , '''9'''  (1992)  pp. 91–109</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Auzinger,   R. Frank,   G. Kirlinger,   "Extending convergence theory for nonlinear stiff problems, Part I"  ''BIT'' , '''36'''  (1996)  pp. 635–652</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Burrage,   J.C. Butcher,   "Stability criteria for implicit Runge–Kutta methods"  ''SIAM J. Numer. Anal.'' , '''16'''  (1979)  pp. 46–57</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.C. Butcher,   "A stability property of implicit Runge–Kutta methods"  ''BIT'' , '''15'''  (1975)  pp. 358–361</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Crouzeix,   "Sur la B-stabilité des méthodes de Runge–Kutta"  ''Numer. Math.'' , '''32'''  (1979)  pp. 75–82</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G. Dahlquist,   "Error analysis for a class of methods for stiff nonlinear initial value problems" , ''Numerical Analysis'' , ''Lecture Notes in Mathematics'' , '''506'''  (1976)  pp. 60–72</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  K. Dekker,   J.G. Verwer,   "Stability of Runge–Kutta methods for stiff nonlinear differential equations" , North-Holland  (1984)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R. Frank,   J. Schneid,   C.W. Ueberhuber,   "The concept of B-convergence"  ''SIAM J. Numer. Anal.'' , '''18'''  (1981)  pp. 753–780</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  R. Frank,   J. Schneid,   C.W. Ueberhuber,   "Stability properties of implicit Runge–Kutta methods"  ''SIAM J. Numer. Anal.'' , '''22'''  (1985)  pp. 497–515</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  R. Frank,   J. Schneid,   C.W. Ueberhuber,   "Order results for implicit Runge–Kutta methods applied to stiff systems"  ''SIAM J. Numer. Anal.'' , '''22'''  (1985)  pp. 515–534</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  E. Hairer,   G. Wanner,   "Solving ordinary differential equations" , '''II: stiff and differential-algebraic problems''' , Springer  (1991)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  A. Prothero,   A. Robinson,   "On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations"  ''Math. Comp.'' , '''28'''  (1974)  pp. 145–162</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  J. Schneid,   "B-convergence of Lobatto IIIC formulas"  ''Numer. Math.'' , '''51'''  (1987)  pp. 229–235</TD></TR></table>