This paper develops the stable adaptive time-dependent mesh scheme for a
one-dimensional linear advection-diffusion equation with homogeneous
Dirichlet boundary conditions and a sinusoidal initial condition. The
aim is to present accurate stable moving nodes finite difference scheme
with its stability and convergence. The boundary layer of the flow is
exponential therefore difference scheme needs mesh refinement. The
moving mesh method analyzes the problem physics and adjusts the mesh
according to the problem as it moves nodes in the region of edges. We
develop numerical results using four MMPDEs with varying numbers of
nodes. A conservative semi-discretization finite difference scheme is
used for the spatial derivative and backward Euler difference scheme is
employed for the temporal derivative. We have presented five cases in
detail to understand the physics of the problem. The proposed moving
mesh finite difference method is considerably more efficient than the
numerical methods offered in the literature.