Bridging the Multiscale Hybrid-Mixed and Multiscale Hybrid High-Order methods

We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) and the Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusion problem with piecewise polynomial source term. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, we prove that the equivalence holds for general polytopal (coarse) meshes and arbitrary approximation orders. We also leverage the interchange of properties to perform a unified convergence analysis, as well as to improve on both methods.

Solving 3-D Gray–Scott Systems with Variable Diffusion Coefficients on Surfaces by Closest Point Method with RBF-FD

The Gray–Scott (GS) model is a non-linear system of equations generally adopted to describe reaction–diffusion dynamics. In this paper, we discuss a numerical scheme for solving the GS system. The diffusion coefficients of the model are on surfaces and they depend on space and time. In this regard, we first adopt an implicit difference stepping method to semi-discretize the model in the time direction. Then, we implement a hybrid advanced meshless method for model discretization. In this way, we solve the GS problem with a radial basis function–finite difference (RBF-FD) algorithm combined with the closest point method (CPM). Moreover, we design a predictor–corrector algorithm to deal with the non-linear terms of this dynamic. In a practical example, we show the spot and stripe patterns with a given initial condition. Finally, we experimentally prove that the presented method provides benefits in terms of accuracy and performance for the GS system’s numerical solution.

Chebyshev collocation method for the variable-order fractional diffusion equation with a variable diffusion coefficient

In this paper, we study the space-time variable-order fractional diffusion equation with a variable diffusion coefficient. The fractional derivatives of variable-orders are considered in the Caputo sense. We propose a numerically efficient pseudospectral method with Chebyshev polynomial as an orthogonal basis function. Also, we examine the error analysis of the given numerical approach. A variation on the maximum absolute error with the different variable orders in space and time are studied. Some illustrative examples are presented with different boundary conditions, e.g., Dirichlet, mixed, and non-local. The applicability of the method is also tested with the problem that has fractional power in solution. The results obtained from the proposed method prove the efficacy and reliability of the method.

Damped Klein-Gordon equation with variable diffusion coefficient

<p style='text-indent:20px;'>We consider a damped Klein-Gordon equation with a variable diffusion coefficient. This problem is challenging because of the equation's unbounded nonlinearity. First, we study the nonlinearity's continuity properties. Then the existence and the uniqueness of the solutions is established. The main result is the continuity of the solution map on the set of admissible parameters. Its application to the parameter identification problem is considered.</p>

Higher-Order Compact Finite Difference for Certain PDEs in Arbitrary Dimensions

In this paper, we first present the expression of a model of a fourth-order compact finite difference (CFD) scheme for the convection diffusion equation with variable convection coefficient. Then, we also obtain the fourth-order CFD schemes of the diffusion equation with variable diffusion coefficients. In addition, a fine description of the sixth-order CFD schemes is also developed for equations with constant coefficients, which is used to discuss certain partial differential equations (PDEs) with arbitrary dimensions. In this paper, various ways of numerical test calculations are prepared to evaluate performance of the fourth-order CFD and sixth-order CFD schemes, respectively, and the empirical results are proved to verify the effectiveness of the schemes in this paper.

Assessment of turbulence effects on effective solute diffusivity close to a sediment-free fluid interface

Our work is focused on the analysis of solute mixing under the influence of turbulent flow propagating in a porous system across the interface with a free fluid. Such a scenario is representative of solute transport and chemical mixing in the hyporheic zone. The study is motivated by recent experimental results (Chandler et al. Water Res Res 52(5):3493–3509, 2016) which suggested that the effective diffusion parameter is characterized by an exponentially decreasing trend with depth below the sediment-water interface. This result has been recently employed to model numerically downstream solute transport and mixing in streams. Our study provides a quantification of the uncertainty associated with the interpretation of the available experimental data. Our probabilistic analysis relies on a Bayesian inverse modeling approach implemented through an acceptance/rejection algorithm. The stochastic inversion workflow yields depth-resolved posterior (i.e., conditional on solute breakthrough data) probability distributions of the effective diffusion coefficient and enables one to assess the impact on these of (a) the characteristic grain size of the solid matrix associated with the porous medium and (b) the turbulence level at the water-sediment interface. Our results provide quantitative estimates of the uncertainty associated with spatially variable diffusion coefficients. Finally, we discuss possible limitations about the generality of the conclusions one can draw from the considered dataset.