  We study the stability properties of explicit marching schemes for
second-kind 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 finite difference or finite element schemes for the heat equation
are stable only if the time step $\Delta t$ is of the order $O(\Delta x^2)$,
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 $c_dT^{d/2}$ times the
norm of the right hand side. For the Robin problem on the half space in any
dimension, with constant Robin (heat transfer) coefficient $\kappa$, we exhibit
a constant $C$ such that the forward Euler scheme is stable if $\Delta t <
C/\kappa^2$, 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 $L^\infty$-norm.
