Do we need decay-preserving error estimate for solving parabolic equations with initial singularity?
CoRR(2024)
摘要
Solutions exhibiting weak initial singularities arise in various equations,
including diffusion and subdiffusion equations. When employing the well-known
L1 scheme to solve subdiffusion equations with weak singularities, numerical
simulations reveal that this scheme exhibits varying convergence rates for
different choices of model parameters (i.e., domain size, final time T, and
reaction coefficient κ). This elusive phenomenon is not unique to the L1
scheme but is also observed in other numerical methods for reaction-diffusion
equations such as the backward Euler (IE) scheme, Crank-Nicolson (C-N) scheme,
and two-step backward differentiation formula (BDF2) scheme. The existing
literature lacks an explanation for the existence of two different convergence
regimes, which has puzzled us for a long while and motivated us to study this
inconsistency between the standard convergence theory and numerical
experiences. In this paper, we provide a general methodology to systematically
obtain error estimates that incorporate the exponential decaying feature of the
solution. We term this novel error estimate the `decay-preserving error
estimate' and apply it to the aforementioned IE, C-N, and BDF2 schemes. Our
decay-preserving error estimate consists of a low-order term with an
exponential coefficient and a high-order term with an algebraic coefficient,
both of which depend on the model parameters. Our estimates reveal that the
varying convergence rates are caused by a trade-off between these two
components in different model parameter regimes. By considering the model
parameters, we capture different states of the convergence rate that
traditional error estimates fail to explain. This approach retains more
properties of the continuous solution. We validate our analysis with numerical
results.
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