Acta Applicandae Mathematicae

, Volume 153, Issue 1, pp 101–124 | Cite as

ℒ-Splines and Viscosity Limits for Well-Balanced Schemes Acting on Linear Parabolic Equations



Well-balanced schemes, nowadays mostly developed for both hyperbolic and kinetic equations, are extended in order to handle linear parabolic equations, too. By considering the variational solution of the resulting stationary boundary-value problem, a simple criterion of uniqueness is singled out: the \(C^{1}\) regularity at all knots of the computational grid. Being easy to convert into a finite-difference scheme, a well-balanced discretization is deduced by defining the discrete time-derivative as the defect of \(C^{1}\) regularity at each node. This meets with schemes formerly introduced in the literature relying on so-called ℒ-spline interpolation of discrete values. Various monotonicity, consistency and asymptotic-preserving properties are established, especially in the under-resolved vanishing viscosity limit. Practical experiments illustrate the outcome of such numerical methods.


Constant/Line Perturbation method (C/L-PM) Fundamental system of solutions ℒ-spline Monotone well-balanced scheme Parabolic Cylinder functions (PCF) Vanishing viscosity 

Mathematics Subject Classification (2010)

65M06 34D15 76M45 76R50 


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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Istituto per le Applicazioni del CalcoloRomeItaly

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