Gauss–Runge–Kutta time discretization of wave equations on evolving surfaces

This paper is concerned with an error analysis for a full discretization of the linear wave equation on a moving surface. The equation is discretized in space by the evolving surface finite element method. Discretization in time is done by Gauß–Runge–Kutta (GRK) methods, aiming for higher-order accuracy in time and unconditional stability of the fully discrete scheme. The latter is established in the natural time-dependent norms by using the algebraic stability and the coercivity property of the GRK methods together with the properties of the spatial semi-discretization. Under sufficient regularity conditions, optimal-order error estimates for this class of fully discrete methods are shown. Numerical experiments are presented to confirm some of the theoretical results.

[1]  David Cohen.  Trigonometric schemes for stiff second-order SDEs , 2006 .

[2]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[3]  Christian Lubich,et al.  Variational discretization of wave equations on evolving surfaces , 2014, Math. Comput..

[4]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[5]  F. Mémoli,et al.  Implicit brain imaging , 2004, NeuroImage.

[6]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[7]  Charles M. Elliott,et al.  Numerical diffusion-induced grain boundary motion , 2001 .

[8]  Alexander Ostermann,et al.  Interior estimates for time discretizations of parabolic equations , 1995 .

[9]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[10]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[11]  Chandrasekhar Venkataraman,et al.  Backward difference time discretization of parabolic differential equations on evolving surfaces , 2013 .

[12]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[13]  Willem Hundsdorfer,et al.  A study of B-convergence of Runge-Kutta methods , 1986, Computing.

[14]  Andreas Dedner,et al.  A generic interface for parallel and adaptive discretization schemes: abstraction principles and the Dune-Fem module , 2010, Computing.

[15]  Charles M. Elliott,et al.  Finite elements on evolving surfaces , 2007 .

[16]  J. M. Sanz-Serna,et al.  Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations , 1987 .

[17]  Martin Rumpf,et al.  A Convergent Finite Volume Scheme for Diffusion on Evolving Surfaces , 2011, SIAM J. Numer. Anal..

[18]  C. Lubich.  VARIATIONAL DISCRETIZATION OF LINEAR WAVE EQUATIONS ON EVOLVING SURFACES , 2012 .

[19]  Charles M. Elliott,et al.  L2-estimates for the evolving surface finite element method , 2012, Math. Comput..

[20]  Gerhard Dziuk,et al.  Runge–Kutta time discretization of parabolic differential equations on evolving surfaces , 2012 .

[21]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[22]  C. M. Elliott,et al.  A Fully Discrete Evolving Surface Finite Element Method , 2012, SIAM J. Numer. Anal..

[23]  E. Hairer,et al.  Stiff and differential-algebraic problems , 1991 .

[24]  O E Jensen,et al.  A theoretical study of surfactant and liquid delivery into the lung. , 1998, Journal of applied physiology.

[25]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[26]  K. Dekker,et al.  The order of B-convergence of the Gaussian Runge-Kutta method , 1986, Computing.

[27]  Alexander Ostermann,et al.  RUNGE-KUTTA METHODS FOR PARABOLIC EQUATIONS AND CONVOLUTION QUADRATURE , 1993 .

[28]  J. Kraaijevanger,et al.  B-convergence of the implicit midpoint rule and the trapezoidal rule , 1985 .

[29]  L. Einkemmer.  Structure preserving numerical methods for the Vlasov equation , 2016, 1604.02616.

[30]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[31]  J. Lowengrub,et al.  A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant , 2004 .