We study the stability properties of explicit marching schemes for secondkind Volterra
integral equations that arise when solving boundary value problems for the heat
equation by means of potential theory. It is well known that explicit finitedifference
or finiteelement schemes for the heat equation are stable only if the time step
$\Delta t$ is of the
order
$\mathcal{O}\left(\Delta {x}^{2}\right)$, where
$\Delta x$ is the finest
spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all
dimensions
$d\ge 1$,
we show that the simplest Volterra marching scheme, i.e., the forward Euler
scheme, is
unconditionally stable. Our proof is based on an explicit spectral
radius bound of the marching matrix, leading to an estimate that an
${L}^{2}$norm
of the solution to the integral equation is bounded by
${c}_{d}{T}^{d\u22152}$
times the norm of the righthand side. For the Robin problem on
the halfspace in any dimension, with constant Robin (heat transfer)
coefficient $\kappa $, we
exhibit a constant
$C$
such that the forward Euler scheme is stable if
$\Delta t<C\u2215{\kappa}^{2}$,
independent of any spatial discretization. This relies on new lower bounds on the
spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally,
we show that the forward Euler scheme is unconditionally stable for the
Dirichlet problem on any smooth
convex domain in any dimension, in the
${L}^{\infty}$norm.
