The Merriman-Bence-Osher threshold dynamics method is an efficient algorithm to simulate the motion by mean curvature. It has the advantages of being easy to implement and with high efficiency. In this paper, we propose a threshold dynamics method for dislocation dynamics in a slip plane, in which the spatial operator is essentially an anisotropic fractional Laplacian. We show that this threshold dislocation dynamics method is able to give two correct leading orders in dislocation velocity, including both the $\mathcal{O}(log ε)$ local curvature force and the $\mathcal{O}(1)$ nonlocal force due to the long-range stress field generated by the dislocations as well as the force due to the applied stress, where $ε$ is the dislocation core size, if the time step is set to be $∆t = ε.$ This generalizes the available result of threshold dynamics with the corresponding fractional Laplacian, which is on the leading order $\mathcal{O}(log \Delta t)$ local curvature velocity under the isotropic kernel. We also propose a numerical method based on spatial variable stretching to correct the mobility and to rescale the velocity for efficient and accurate simulations, which can be applied generally to any threshold dynamics method. We validate the proposed threshold dislocation dynamics method by numerical simulations of various motions and interaction of dislocations.

In this paper we develop and analyze two energy-preserving hybrid asymptotic augmented finite volume methods on uniform grids for nonlinear weakly degenerate and strongly degenerate wave equations. In order to deal with the degeneracy, we introduce an intermediate point to divide the whole domain into singular subdomain and regular subdomain. Then Puiseux series asymptotic technique is used in singular subdomain and augmented finite volume scheme is used in regular subdomain. The keys of the method are the recovery of Puiseux series in singular subdomain and the appropriate combination of singular and regular subdomain by means of augmented variables associated with the singularity. Although the effect of singularity on the calculation domain is conquered by the Puiseux series reconstruction technique, it also brings difficulties to the theoretical analysis. Based on the idea of staggered grid, we overcome the difficulties arising from the augmented variables related to singularity for the construction of conservation scheme. The discrete energy conservation and convergence of the two energy-preserving methods are demonstrated successfully. The advantages of the proposed methods are the energy conservation and the global convergence order determined by the regular subdomain scheme. Numerical examples on weakly degenerate and strongly degenerate under different nonlinear functions are provided to demonstrate the validity and conservation of the proposed method. Specially, the conservation of discrete energy is also ensured by using the proposed methods for both the generalized Sine-Gordeon equation and the coefficient blow-up problem.

We present a second-order strictly length-preserving and unconditionally energy-stable rotational discrete gradient (Rdg) scheme for the numerical approximation of the Oseen-Frank gradient flows with anisotropic elastic energy functional. Two essential ingredients of the Rdg method are reformulation of the length constrained gradient flow into an unconstrained rotational form and discrete gradient discretization for the energy variation. Besides the well-known mean-value and Gonzalez discrete gradients, we propose a novel Oseen-Frank discrete gradient, specifically designed for the solution of Oseen-Frank gradient flow. We prove that the proposed Oseen-Frank discrete gradient satisfies the energy difference relation, thus the resultant Rdg scheme is energy stable. Numerical experiments demonstrate the efficiency and accuracy of the proposed Rdg method and its capability for providing reliable simulation results with highly disparate elastic coefficients.

In the numerical simulation of compressible turbulence involving shock waves, accurately capturing the intricate vortex structures and robustly computing the shock wave are imperative. Employing a high-order scheme with adaptive dissipation characteristics proves to be an efficient approach in distinguishing small-scale vortex structures with precision while capturing discontinuities. However, differentiating between small-scale vortex structures and discontinuities during calculations has been a key challenge. This paper introduces a high-order adaptive dissipation central-upwind weighted compact nonlinear scheme based on vortex recognition (named as WCNS-CU-Ω), that is capable of physically distinguishing shock waves and small-scale vortex structures in the high wave number region by identifying vortices within the flow field, thereby enabling adaptive control of numerical dissipation for interpolation schemes. A variety of cases involving Euler, N-S even RANS equations are tested to verify the performance of the WCNS-CU-Ω scheme. It was found that this new scheme exhibits excellent small-scale resolution and robustness in capturing shock waves. As a result, it can be applied more broadly to numerical simulations of compressible turbulence.

One of the ways natural and synthetic systems regulate temperature is via circulating fluids through vasculatures embedded within their bodies. Because of the flexibility and availability of proven fabrication techniques, vascular-based thermal regulation is attractive for thin microvascular systems. Although preliminary designs and experiments demonstrate the feasibility of thermal modulation by pushing fluid through embedded micro-vasculatures, one has yet to optimize the performance before translating the concept into real-world applications. It will be beneficial to know how two vital design variables—host material’s thermal conductivity and fluid’s heat capacity rate—affect a thermal regulation system’s performance, quantified in terms of the mean surface temperature. This paper fills the remarked inadequacy by performing adjoint-based sensitivity analysis and unravels a surprising non-monotonic trend. Increasing thermal conductivity can either increase or decrease the mean surface temperature; the increase happens if countercurrent heat exchange—transfer of heat from one segment of the vasculature to another—is significant. In contrast, increasing the heat capacity rate will invariably lower the mean surface temperature, for which we provide mathematical proof. The reported results (a) dispose of some misunderstandings in the literature, especially on the effect of the host material’s thermal conductivity, (b) reveal the role of countercurrent heat exchange in altering the effects of design variables, and (c) guide designers to realize efficient microvascular active-cooling systems. The analysis and findings will advance the field of thermal regulation both on theoretical and practical fronts.

In this paper, we propose a conservative and positivity-preserving method to solve the anisotropic diffusion equations with the physics-informed neural network (PINN). Due to the possible complicated discontinuity of diffusion coefficients, without employing multiple neural networks, we approximate the solution and its gradients by one single neural network with a novel first-order loss formulation. It is proven that the learned solution with this loss formulation only has the $\mathcal{O}(\varepsilon)$ flux conservation error theoretically, where the parameter $\varepsilon$ is small and user-defined, while the loss formulation with the original PDE with/without flux conservation constraints may have $\mathcal{O}(1)$ flux conservation error. To keep positivity with the neural network approximation, some positive functions are applied to the primal neural network solution. This loss formulation with some observation data can also be employed to identify the unknown discontinuous coefficients. Compared with the usual PINN even with the direct flux conservation constraints, it is shown that our method can significantly improve the solution accuracy due to the better flux conservation property, and indeed preserve the positivity strictly for the forward problems. It can predict the discontinuous diffusion coefficients accurately in the inverse problems setting.

In this paper, we construct a family of temporal high-order parametric relaxation Runge–Kutta (pRRK) schemes for stiff ordinary differential equations (ODEs), and explore their application in solving hyperbolic conservation laws with source terms. The new time discretization methods are explicit, large time-stepping, delay-free and able to preserve steady state. They are combined with fifth-order weighted compact nonlinear schemes (WCNS5) spatial discretization and parametrized maximum-principle-preserving (MPP) flux limiters to solve scalar hyperbolic equations with source terms. We prove that the fully discrete schemes preserve the maximum principle strictly. Through benchmark test problems, we demonstrate that the proposed schemes have fifth-order accuracy in space, fourth-order accuracy in time and allow for large time-stepping without time delay. Both theoretical analyses and numerical experiments are presented to validate the benefits of the proposed schemes.

We construct a positivity-preserving and well-balanced high order accurate finite difference scheme for solving shallow water equations under the fourth order compact finite difference framework. The source term is rewritten to balance the flux gradient in steady state solutions. Under a suitable CFL condition, the proposed compact difference scheme satisfies weak monotonicity, i.e., the average water height defined by the weighted average of a three-points stencil stays non-negative in forward Euler time discretization. Thus, a positivity-preserving limiter can be used to enforce the positivity of water height point values in a high order strong stability preserving Runge-Kutta method. A TVB limiter for compact finite difference scheme is also used to reduce numerical oscillations, without affecting well-balancedness and positivity. Numerical experiments verify that the proposed scheme is high-order accurate, positivity-preserving, well-balanced and free of numerical oscillations.