Abstract
Abstract
We develop a discontinuous Galerkin method for advection--diffusion equations with a Caputo derivative whose order is evaluated at the current observation time. Backward-Euler generalized convolution quadrature is represented by a positive integral of discrete resolvent products on arbitrary time grids. Composite Gaussian quadrature compresses this integral into auxiliary states that are independent of the fractional order, while the current increment coefficient is retained exactly. Complete monotonicity ensures a nonnegative diagonal correction and preserves the comparison structure of the discrete memory operator. Combined with symmetric interior penalty diffusion and upwind convection, this structure yields $L^2$-norm stability without restrictions on adjacent-step ratios or the variation of the sampled order. For smooth solutions, we prove first-order temporal convergence, including a nonzero initial slope, and optimal spatial $L^2$ convergence under explicit coefficient and adjoint-regularity assumptions. Scaled scalar derivative states provide computable bounds for the Gaussian remainder and the omitted tails. Together with the algebraic residual, these bounds control the discrepancy from the scheme with exact integral weights. Two-dimensional experiments with anisotropic, time-dependent diffusion examine temporal and spatial convergence, weak initial singularities, compression errors, and stability on highly irregular grids. Comparisons with the nonuniform L1 formula quantify the accuracy tradeoff of the first-order CQ discretization.
MSC 2020: 65M12, 65M15, 65M60, 26A33.