Runge-Kutta time discretization of parabolic differential equations on evolving surfaces (Q2882350)

Under rather general hypotheses, conservation of a scalar quantity \(u(x,t)\) with a linear diffusive flux on a smoothly evolving family of smooth hypersurfaces \(\Gamma(t)\) with velocity \(v\) can be modeled by a linear parabolic partial differential equation with a certain initial condition on \(\Gamma(0)\). It is therefore of interest to construct efficient methods to discretize this class of partial differential equations. The process involves two stages. In the first one, an evolving surface finite element method based on piecewise linear finite elements is applied. The moving surface \(\Gamma(t)\) is approximated by a moving discrete surface \(\Gamma_h(t)\). This results in a large ordinary differential equation system of the form NEWLINE\[NEWLINE \frac{d}{dt} (M(t) u(t)) + A(t) u(t) = f(t), \qquad u(0)=u_0, \tag{1}NEWLINE\]NEWLINE where \(M(t)\) (the mass matrix) is symmetric positive-definite, \(A(t)\) (the stiffness matrix) is symmetric and positive semidefinite (since only closed surfaces are considered) and \(u(t) \in \mathbb{R}^n\) represents the discrete solution \(u_h(x,t)\). Both matrices \(M\) and \(A\) are sparse.NEWLINENEWLINEIn the second stage, the system (1) is rewritten as NEWLINE\[NEWLINE \begin{aligned} \frac{dy}{dt}(t) + A(t) u(t) & = f(t) \\ y(t)& = M(t) u(t)\end{aligned} \tag{2}NEWLINE\]NEWLINE to which an \(m\)-stage implicit Runge-Kutta method of the stage order \(q \geq 1\) and the classical order \(p \geq q\) is applied in the standard way as for differential-algebraic equations. If the Runge-Kutta method is assumed to be algebraically stable and stiffly accurate then unconditional stability of the full discretization process is proven and the convergence properties are analyzed. The implementation details are given in particular for the Radau IIA method and two specific examples are provided to illustrate the main theoretical results.