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Patterns arising from the reaction–diffusion system (1) on a stationary cylinder correspond to the u chemical concentration. Stripe patterns emerge which evolve into spot patterns as time increases. Parameter values are given in Table 2 (Experiment 5). Observe how patterns are non-uniformly distributed and seem to depend on the curvature which is non-uniform across the cylinder. (Color version online.)
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In this paper we present a robust, efficient and accurate finite element method for solving reaction-diffusion systems on stationary spheroidal surfaces (these are surfaces which are deformations of the sphere such as ellipsoids, dumbbells, and heart-shaped surfaces) with potential applications in the areas of developmental biology, cancer research...
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... Therefore, it becomes attractive to explore numerical methods to understand the dynamical behaviour of RDSs. It has been well demonstrated in the literature that the finite element method is one of the powerful numerical method for the spatial discretization of RDSs [31,32,33,34,35]. According to this procedure, the system of partial differential equations is first discretized in space using the finite element method which produces a time-dependent system of ordinary differential equations (ODEs), followed by the time discretization using several time-stepping schemes to obtain the desired numerical solution [36]. ...
In this work, domain-dependent stability analysis is conducted for a reaction diffusion system with linear cross-diffusion. Its spatiotemporal dynamics are explored for the case of an activator-depleted model of two chemical species in terms of the domain size and its model parameters. Linear stability analysis is employed to derive the constraints which are necessary in understanding the role of linear cross-diffusion in driving the instability of the system. The conditions are proven in terms of lower and upper bounds of the domain size together with the reaction, self and cross-diffusion coefficients. The full parameter classification of the model system is presented in terms of the relationship between the domain size and cross-diffusion-driven instability. Subsequently, regions showing Turing instability, Hopf and transcritical types of bifurcations are demonstrated using the parameter values of the system. Theoretical findings are supported by the finite element simulations on two dimensional rectangular domain. Numerical simulations showing each of the three types of dynamics are examined for the Schnakenberg kinetics, also known as an activator-depleted reaction kinetics.
... On two-dimensional spatial domains, both mechanisms are able to create a wide variety of patterns, including spots, stripes, and gaps, depending on the precise structure of the model, the parameter values, and the shape and size of the domain. [50][51][52]. ...
... In [21], a multilevel finite element approach with spatial and temporal adaptivity was constructed for reaction-diffusion problems. In [22], a projected finite element approach on stationary closed surface geometries, together with a backward Euler time integration, was used to simulate pattern-formation in biological applications. In [23], a moving mesh finite element method was constructed for simulating chemotaxis in two-dimensions. ...
A new class of implicit-explicit (IMEX) methods combined with a p-adaptive mixed finite element formulation is proposed to simulate the diffusion of reacting species. Hierarchical polynomial functions are used to construct an $H(\mathrm{Div})$-conforming base for the flux vectors, and a non-conforming $L^2$ base for the mass concentration of the species. The mixed formulation captures the distinct nonlinearities associated with the constitutive flux equations and the reaction terms. The IMEX method conveniently treats these two sources of nonlinearity implicitly and explicitly, respectively, within a single time-stepping framework. The combination of the p-adaptive mixed formulation and the IMEX method delivers a robust and efficient algorithm. The proposed methods eliminate the coupled effect of mesh size and time step on the algorithmic stability. A residual based a posteriori error estimate that provides an upper bound of the natural error norm is derived. The availability of such estimate which can be obtained with minimal computational effort and the hierarchical construction of the finite element spaces allow for the formulation of an efficient p-adaptive algorithm. A series of numerical examples demonstrate the performance of the approach. It is shown that the method with the p-adaptive strategy accurately solves problems involving travelling waves, and those with discontinuities and singularities. The flexibility of the formulation is also illustrated via selected applications in pattern formation and electrophysiology.
... Lemma 3.1 [54]. (The locally Lipschitz) Let (u 1 , v 1 ) and (u 2 , v 2 ) be solutions of (1.1) for initial conditions (u 1, 0 , v 1, 0 ) ∈ Ω and (u 2, 0 , v 2, 0 ) ∈ Ω respectively, then: ...
In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2‐D reaction–diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction–diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P0, P0, RT0). The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction–diffusion systems, stability analysis and error bounds for semi‐discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well‐known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.
... The increasing interest from applications in RDSs on manifolds has stimulated the development of several numerical methods for such systems. Among the methods for PDEs on stationary surfaces we recall finite differences (Varea et al., 1999), the spectral method of lines (Chaplain et al., 2001), closest point methods (see Macdonald & Ruuth, 2009 and references therein), kernel methods (see Shankar et al., 2015 and references therein), embedding methods (see Bertalmío et al., 2003), surface finite element methods (SFEM) and their extensions (see Dziuk, 1988;Olshanskii et al., 2009;Dziuk & Elliott, 2013a;Burman et al., 2015;Tuncer et al., 2015). In this article, we consider a lumped mass surface finite element method (LSFEM) for the spatial discretization of Eqs. ...
We propose and analyse a lumped surface finite element method for the numerical approximation of
reaction–diffusion systems on stationary compact surfaces in R3. The proposed method preserves the
invariant regions of the continuous problem under discretization and, in the special case of scalar equations,
it preserves the maximum principle. On the application of a fully discrete scheme using the implicit–explicit
Euler method in time, we prove that invariant regions of the continuous problem are preserved (i) at the
spatially discrete level with no restriction on the meshsize and (ii) at the fully discrete level under a timestep
restriction. We further prove optimal error bounds for the semidiscrete and fully discrete methods, that is,
the convergence rates are quadratic in the meshsize and linear in the timestep. Numerical experiments are
provided to support the theoretical findings.We provide examples in which, in the absence of lumping, the
numerical solution violates the invariant region leading to blow-up.
... 14). Among the various discretisation techniques for surface PDEs existing in literature (see for example [24,26,[33][34][35][36][37][38]) we consider the Surface Finite Element Method (SFEM) introduced in the seminal paper [39]. The core idea is to approximate the surface with a polygonal surface made, as in the planar case, of triangular non-overlapping elements whose vertices belong to the surface and to consider a space of piecewise linear functions. ...
We present and analyze a Virtual Element Method (VEM) of arbitrary polynomial order $k\in\mathbb{N}$ for the Laplace-Beltrami equation on a surface in $\mathbb{R}^3$. The method combines the Surface Finite Element Method (SFEM) [Dziuk, Elliott, \emph{Finite element methods for surface PDEs}, 2013] and the recent VEM [Beirao da Veiga et al, \emph{Basic principles of Virtual Element Methods}, 2013] in order to handle arbitrary polygonal and/or nonconforming meshes. We account for the error arising from the geometry approximation and extend to surfaces the error estimates for the interpolation and projection in the virtual element function space. In the case $k=1$ of linear Virtual Elements, we prove an optimal $H^1$ error estimate for the numerical method. The presented method has the capability of handling the typically nonconforming meshes that arise when two ore more meshes are pasted along a straight line. Numerical experiments are provided to confirm the convergence result and to show an application of mesh pasting.
... That is, initially the source terms are spatially uniform. Then Eq. (18) and (19) are evaluated at ...
... Then Eq. (18) and (19) are evaluated at N = 2 and k = 1, 2: ...
... This procedure is repeated for all successive time steps, with three spatial sums needed at N = 3, four spatial sums needed at N = 4, and so on as required by the k-summation in Eq. (18) and (19). This behavior arises because the summation over time index k is a convolution sum. ...
Estimation of thermal properties, diffusion properties, or chemical–reaction rates from transient data requires that a model is available that is physically meaningful and suitably precise. The model must also produce numerical values rapidly enough to accommodate iterative regression, inverse methods, or other estimation procedures during which the model is evaluated again and again. Applications that motivate the present work include process control of microreactors, measurement of diffusion properties in microfuel cells, and measurement of reaction kinetics in biological systems. This study introduces a solution method for nonisothermal reaction–diffusion (RD) problems that provides numerical results at high precision and low computation time, especially for calculations of a repetitive nature. Here, the coupled heat and mass balance equations are solved by treating the coupling terms as source terms, so that the solution for concentration and temperature may be cast as integral equations using Green’s functions (GF). This new method requires far fewer discretization elements in space and time than fully numeric methods at comparable accuracy. The method is validated by comparison with a benchmark heat transfer solution and a commercial code. Results are presented for a first-order chemical reaction that represents synthesis of vinyl chloride.
... Although this is a widely acknowledged motif the systematic numerical analysis of reaction-diffusion systems that exhibit this kind of coupling is in my eyes still lacking behind surprisingly. Related work using similar coupling schemes between a passive bulk and reactive membrane as catalytic templates can be found in the work by Gomez-Marin et al. [169] for nonlinear Fitzhugh-Nagumo dynamics leading to self-sustained spatiotemporal dynamics, by Rätz et al. for Turing instabilities and symmetry breaking in signaling networks [170][171][172], bulk-surface coupled Cahn-Hilliard systems [173], receptor-ligand dynamics [174] and for generic bulk-surface PDE problems as shown in [168,[175][176][177][178]]. ...
As the life sciences become more quantitative, particle-based simulation tools can be used to model the complex spatiotemporal dynamics of biological systems with single particle resolution. In particular, they naturally account for the stochastic nature of molecular reactions. Here I apply this approach to three different biological systems that are intrinsically stochastic. As an example for cellular information processing, we investigate the receptor dynamics of the interferon type I system and show that asymmetric dimerization reactions between signaling receptors in the plasma membrane enable cells to discriminate between different ligands. Using an information theoretic framework, we show why the binding asymmetry enables this system to become robust against ligand concentration fluctuations. As an example for structure formation, we analyze the role of stochasticity and geometrical confinement for the Min oscillations that bacteria use to determine their middle. We predict mode selection as a function of geometry in excellent agreement with recent experiments and quantify the stochastic switching of oscillation modes leading to multistable oscillation patterns. As an ex- ample for self-assembly, we use a multiparticle collision dynamics (MPCD) approach to address how shear flow modulates the assembly of rings and capsids. We find that an intermediate level of shear flow can help to suppress the emergence of malformed structures. Together, these projects demonstrate the power and wide applicability of particle-based computer simulations of biological reaction-diffusion systems.
... The PFEM proposed in this article is inspired by the radially projected finite element method which was used to compute approximate numerical solutions for partial differential equations on stationary spheroidal surfaces such as spheres, ellipsoids, and tori [29,36,37]. The PFEM gives a geometrically exact discretization of spheroidal surfaces (the geometry is not approximated, but represented exactly) and is attractive for numerical simulations since the resulting finite element discretization is conforming and is "logically rectangular." ...
... First, we introduce the projection operator, P, on closed stationary surfaces which was introduced in [37] for the case of projected finite elements. Next, we describe how this projection can be generalized to take into account surface evolution. ...
... Remark 2.3. The quality of the mesh generated by the Lipschitz continuous mappings P(x) have been studied in [29,37]. Since for any fixed time, the time-dependent projection operators P(x, t) and P −1 (x, t) are Lipschitz continuous, the meshes generated by these projectors are also shape regular. ...
The focus of this article is to present the projected finite element method for solving systems of reaction-diffusion equations on evolving closed spheroidal surfaces with applications to pattern formation. The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is “logically” rectangular. Furthermore, the surface is not approximated but described exactly through the projection. The surface evolution law is incorporated into the projection operator resulting in a time-dependent operator. The time-dependent projection operator is composed of the radial projection with a Lipschitz continuous mapping. The projection operator is used to generate the surface mesh whose connectivity remains constant during the evolution of the surface. To illustrate the methodology several numerical experiments are exhibited for different surface evolution laws such as uniform isotropic (linear, logistic and exponential), anisotropic, and concentration-driven. This numerical methodology allows us to study new reaction-kinetics that only give rise to patterning in the presence of surface evolution such as the activator-activator and short-range inhibition; long-range activation .
... For example, the ESFEM method has been applied in [2] to solve the well-known Schnakenberg reaction-diffusion system on evolving surfaces. Other numerical approaches to solve reaction-diffusion systems on static surfaces are given, for example, in [16,63,68,69]. For our purposes, here we describe the Lumped SFEM method in the case of a general C 2 compact surface Γ in R 3 without boundary, i.e. ...
The present paper deals with the pattern formation properties of a specific morpho-electrochemical reaction-diffusion model on a sphere. The physico-chemical background to this study is the morphological control of material electrodeposited onto spherical particles. The particular experimental case of interest refers to the optimisation of novel metal-air flow batteries and addresses the electrodeposition of zinc onto inert spherical supports. Morphological control in this step of the high-energy battery operation is crucial to the energetic efficiency of the recharge process and to the durability of the whole energy-storage device. To rationalise this technological challenge within a mathematical modelling perspective, we consider the reaction-diffusion system for metal electrodeposition introduced in [Bozzini et al., J. Solid State Electr.17, 467–479 (2013)] and extend its study to spherical domains. Conditions are derived for the occurrence of the Turing instability phenomenon and the steady patterns emerging at the onset of Turing instability are investigated. The reaction-diffusion system on spherical domains is solved numerically by means of the Lumped Surface Finite Element Method (LSFEM) in space combined with the IMEX Euler method in time. The effect on pattern formation of variations in the domain size is investigated both qualitatively, by means of systematic numerical simulations, and quantitatively by introducing suitable indicators that allow to assign each pattern to a given morphological class. An experimental validation of the obtained results is finally presented for the case of zinc electrodeposition from alkaline zincate solutions onto copper spheres.
