Convergence of Runge-Kutta methods applied to linear partial differential-algebraic equations. (English) Zbl 1072.65122

Debrabant, K.; Strehmel, K.

Appl. Numer. Math. 53, No. 2-4, 213-229 (2005).

Linear partial differential-algebraic equations (PDAEs) of the form \[ A u_t(t,x) + B(u_{xx}(t,x) + r u_x(t,x)) + C u(t,x) =f(t,x) \] where \( t \in ( t_0, t_e), \;\;x \in (-l,l)\subset \mathbb R \) and \(A, B, C \in \mathbb R^{n,n}\) are constant matrices, \( r \in \mathbb R, u,f \in [t_0,t_e]\times [-l,l]\rightarrow \mathbb R^n\), and what is more interesting the matrix \(A\) is singular, what leads to the differential-algebraic problem.
In this case it is impossible to prescribe initial and boundary conditions for all components of the solution vector, they must fulfill certain conditions. A practical example how to do it is shown for the so called superconducting coil.
The problem of linear PDAEs is investigated from the numerical point of view. The discretization in space via finite differences is used and then a Runge-Kutta approximation of the method of lines for differential algebraic equations (MOL-DAE) is derived. When \(N\) is the number of space grid points, by a regular transformation the MOL-DAE of dimension \(nN\) is decoupled into \(N\) systems of dimension \(n\) and the Weierstrass-Kronecker transformation for each of these systems into a system of ordinary differential equations and an algebraic system. The so called differential time index of the PDAE is introduced which gives the Runge-Kutta approximation to these subsystems. The convergence of \(L\)-stable Runge-Kutta discretizations with constant step sizes is proved. The obtained order of convergence in time depends on the differential time index of the PDAE and on the boundary condition if they are homogeneous or not.
Included numerical examples confirm the theoretical results for the backward Euler method and 3-stage Radau IIA method.

Reviewer: Angela Handlovičová (Bratislava)

MSC:

65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M20	Method of lines for initial value and initial-boundary value problems involving PDEs
65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
35R10	Partial functional-differential equations
35G15	Boundary value problems for linear higher-order PDEs

Keywords:

partial differential-algebraic equations; coupled systems; implicit Runge-Kutta methods; stability; convergence; differential time index; numerical examples; backward Euler method; 3-stage Radau IIA method

Software:

RODAS

References:

[1]	Brenner, P.; Crouzeix, M.; Thomée, V., Single step methods for inhomogeneous linear differential equations in Banach space, RAIRO anal. numér., 16, 5-26, (1982) · Zbl 0477.65040
[2]	Campbell, S.L.; Marszalek, W., The index of an infinite dimensional implicit system, Math. comput. modelling dynamical systems, 5, 18-42, (1999) · Zbl 0922.35040
[3]	K. Debrabant, Numerische Behandlung linearer und semilinearer partieller differentiell-algebraischer Systeme mit Runge-Kutta-Verfahren, Dissertation, Martin-Luther-Universität Halle-Wittenberg, 2004
[4]	Golub, G.H.; van Loan, C.F., Matrix computations, (1996), The John Hopkins University Press Baltimore, MD · Zbl 0865.65009
[5]	Hairer, E.; Wanner, G., Solving ordinary differential equations II. stiff and differential-algebraic problems, (1996), Springer Berlin · Zbl 0859.65067
[6]	Lubich, Ch.; Ostermann, A., Runge – kutta methods for parabolic equations and convolution quadrature, Math. comp., 60, 105-131, (1993) · Zbl 0795.65062
[7]	Lucht, W.; Strehmel, K.; Eichler-Liebenow, C., Indexes and special discretization methods for linear partial differential algebraic equations, Bit, 39, 484-512, (1999) · Zbl 0941.65096
[8]	Marszalek, W.; Trzaska, Z.W., Analysis of implicit hyperbolic multivariable systems, Appl. math. modelling, 19, 400-410, (1995) · Zbl 0832.65105
[9]	Ostermann, A.; Roche, M., Runge – kutta methods for partial differential equations and fractional order of convergence, Math. comp., 59, 403-420, (1992) · Zbl 0769.65068
[10]	Ostermann, A.; Thalhammer, M., Convergence of runge – kutta methods for nonlinear parabolic equations, Appl. numer. math., 42, 367-380, (2002) · Zbl 1004.65093