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

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.

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

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References:

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