Asymptotically Compatible Variational Energy Dissipation Laws and Structure-Preserving Variable-Step Linear Relaxation Schemes for a Subdiffusive Tumor-Nutrient Model

Abstract

In this work, we develop a mass-conservative subdiffusive tumor-nutrient model coupled through chemotaxis and memory-dependent reaction exchange, together with structure-preserving variable-step linear relaxation schemes. A weighted Riemann-Liouville identity is proved and used to construct a variational energy and the associated energy dissipation law for the proposed model. Both the model and the associated variational energy dissipation law are asymptotically compatible with their integer-order counterparts as the fractional order approaches one. For numerical approximation, we discretize the Riemann-Liouville fractional derivatives using the L1R formula on nonuniform time meshes and derive the associated discrete weighted inequality. We first construct a variable-step linear relaxation scheme for the subdiffusive model. Then, to mitigate alternating perturbations induced by the auxiliary-variable error and amplified by the L1R convolution at small fractional orders, we construct a temporally damped linear relaxation scheme. Since temporal damping leaves the auxiliary-variable recurrence unchanged, we further introduce an a posteriori relaxation correction to reduce alternating errors in the auxiliary variable. All three schemes preserve total mass, are proven to satisfy discrete variational energy dissipation laws using the discrete weighted inequality, and remain consistent with the integer-order limit. Several numerical examples are presented to validate the accuracy and stability of the proposed schemes.