In this paper, on two-dimension unstructured meshes, a fully-discrete scheme is presented for the reaction-diffusion systems, which are often used as mathematical models for many biological, physical and chemical applications. By using local discontinuous Galerkin (LDG) method, the scheme can derive the numerical approximations not only for solutions but also for their gradients at the same time. In addition, the scheme employs the implicit integration factor (IIF) method for temporal discretization, which allows us to take the time-step as δ t=O(h_min) , and can be computed element by element, so that it reduces the computational cost greatly. Numerical simulations for the chlorite-iodide-malonic acid (CIMA) model demonstrate the expected behavior of the solutions, the efficiency and advantages of the proposed scheme.
Evaporation of liquid droplets on solid substrates is used in many technological processes. Complexity and interdependence of the physical processes inside droplet and their mathematical descriptions, phase transitioning, energy and mass balance, points of singularity and thermocappilar effects force researchers to simplify models and to use numerical methods. We derive solution that has not non-singular heat flux at the droplet edge and describes the vapor concentration out of droplet and temperature at the surface. It allowed us to find out the Marangonni force and calculate non-singular velocity field that can change its direction in the stagnation points.
We consider the piecewise linear finite element method in space for solving the one dimensional wave equation on general spatial meshes. The discretization in time is performed using the Newmark method. We show that the error between the finite element approximate solution and the piecewise linear interpolant of the exact solution is of order $$(h+k)^2$$ in several discrete norms. We construct a new approximation of order $$(h+k)^3$$ . This new third order approximation can be computed using the same linear systems used to compute the finite element approximate solution with the same matrices while the right hand sides are changed. The matrices used to compute this new high-order approximation are tridiagonal and consequently the systems involving these matrices are easily to solve. The convergence analysis is performed in several discrete norms.
This paper presents the development of analytic-numerical approaches to study periodically moving fronts in singularly perturbed reaction-diffusion-advection models. We describe the results of rigorous asymptotic treatment of the problem and suggest a method to generate a dynamic adapted mesh for the numerical solution of such problems. This method based on a priori information. In particular, we take into account a priori estimates on the location of the transition layer, its width and structure. An example is presented to demonstrate the effectiveness of the proposed method.
In this paper a fractional heat equation is considered; it has a Caputo time-fractional derivative of order δ where 0<δ <1 . It is solved numerically on a uniform mesh using the classical L1 and standard three-point finite difference approximations for the time and spatial derivatives, respectively. In general the true solution exhibits a layer at the initial time t=0 ; this reduces the global order of convergence of the finite difference method to O(h^2+τ ^δ ) , where h and τ are the mesh widths in space and time, respectively. A new estimate for the L1 approximation shows that its truncation error is smaller away from t=0 . This motivates us to investigate if the finite difference method is more accurate away from t=0 . Numerical experiments with various non-smooth and incompatible initial conditions show that, away from t=0 , one obtains O(h^2+τ ) convergence.
A new variant of differential evolution (DE) algorithm with a selection of mutation strategy based on the mutant point distance (DEMD) is proposed. Three DEMD variants are compared with state-of-the-art DE variants on CEC 2015 problems at four dimension levels. The results show that one of proposed DEMD variants performs best in 35% of the problems compared to the other examined DE algorithms.
The present work is devoted to computational aspects of solving the optimization problems for semi-linear elliptic interface problems. We develop numerical algorithms for minimizing a cost functional, depending on a state of the system and a control. Numerical experiments are included. The results from computer experiment showed the effectiveness of the approximate method of solution.
We describe a finite difference scheme for a multidimensional advection equation with time delay. The difference scheme has the second order in space and the first order in time and is unconditionally stable. The difference scheme lead to a big system of linear algebraic equations which could be solved in parallel. The performance of a sequential algorithm and several parallel implementations with the MPI technology in the C/C++ language has been studied in test examples, strong scalability is closed to ideal one.
The model of heating of surface of solid substance with infra-red laser and cooling with air flow is created. From initial differential partial equations the discrete form of equation is obtained. The method of integration of system under dynamic change of energy of laser beam and constant parameters of airflow is worked out. Due to this method the initial model was transformed into a discrete form. The discrete equation is solved for the case of dynamically changing laser beam energy and constant parameters of the airflow. The result of solving of equation is graphed in 3D space: distance from center of laser beam/distance from surface of target/temperature.
A method for solving linear systems of equations with sign-definite self-adjoint operator matrix by the Richardson iteration method in case of the absence of information about the lower spectral bound of a problem is presented. The algorithm is based on the simultaneous operation of two competing iterative processes, the effectiveness of which is constantly analyzed. The method is explained on an example of one-dimensional steady-state heat equation.
Problem of obtaining data on the properties of gaseous media is considered. Gases of interest are the gases used as transport systems in technical facilities. The focus is on calculating the kinetic coefficients of gaseous medium considering the molecular processes that take place in the gas flow. Molecular dynamics method is selected as the method of modeling. Various techniques for determining the kinetic coefficients of gases are described in detail and compared. The problem is considered on the example of nitrogen flow. For this goal calculating the coefficients of self-diffusion, shear viscosity and thermal conductivity for nitrogen is made. The obtained numerical results are in good agreement with known theoretical estimates and experimental data.
A singularly perturbed initial-boundary value problem for the parabolic reaction-diffusion-advection (RDA) equation is considered. Some effective asymptotic-numerical approach for the description of internal layers location and moving fronts dynamics is proposed. Asymptotic analysis allows to reduce the spatial dimension of the numerical problem and highlight a priori information to optimize numerical calculations and save computational resources. But for some classes of RDA problems, featuring the internal layers or moving fronts, the layers location or fronts speed could not be found explicitly and asymptotic algorithm needs to be supplemented by the appropriate numerical calculations. In this paper we show the main ideas of the asymptotic algorithm for this type of solutions and outline some problems which need to use numerical calculations on some steps of the asymptotic procedure. The main features of numerical algorithm are presented.
The main goal of this paper is to analyze the fan-mechanism of rotational motion transmission in a system of elastically bonded slabs on flat surface, simulating growth of shear ruptures in super brittle rocks. A physical model recently designed demonstrates that the fan-structure formation can be stable at the absence of distributed shear stress applied. The action of distributed shear stress causes the fan propagation as a wave representing the rupture head. The developed mathematical model of a fan-structure as a continuous system establishes the relation between the fan velocity and the fan length. It is shown that in the absence of friction the fan velocity may be arbitrary, but not greater than the limit velocity which is determined by the moment of inertia of slabs, the initial angle of their orientation and the elastic coefficient of bonds. In a system with friction the velocity of traveling fan is solely determined by the opening angle. The action of distributed shear stress leads to the instability start before the fan-structure completion. The fan length decreases with increasing velocity.
We develop a semi-Lagrangian algorithm for solving the three-dimensional advection problem. A numerical solution is determined on a uniform cubic grid as a piecewise trilinear function. The method is based on the integral balance equation between two neighboring time levels. The domain of integration at the previous time level is a curved cuboid. To compute an integral over this domain numerically, we approximate this cuboid by another one with the same 8 vertices. The latter cuboid is obtained by a trilinear (isoparametric) transformation of the unit cube. This leads to the integration over the unit cube with the help of the composite midpoint rule. Such a technique provides the validity of the local balance equation and does not involve computational and algorithmic complexity for solving the three-dimensional problem. The numerical experiments confirm the first-order convergence.
We investigate the one-dimensional continuation problem (the Cauchy problem) for the parabolic equation with the data on the part of the boundary. For numerical solution we apply finite-difference scheme inversion, the singular value decomposition and the gradient method of the minimizing the goal functional. The comparative analysis of numerical methods are presented.
The new method for solving the three-dimensional boundary value problem for the Helmholtz equation, based on decomposition of an exterior domain, is proposed. The approach proposed is based on using of Swartz alternating method with solving sequentially interior and exterior boundary value problems in overlapping subdomains with iterated interface conditions on their adjacent borders. The sufficient conditions of the method convergence in the case of a negative coefficient in the Helmholtz equation are found. The approach in question is implemented to solve a problem with a complex configuration of boundaries with using the finite volume method for solving the interior boundary value problems and Greens formula for solving the exterior boundary value problems. The rate of convergence of iterations and achievable accuracy of calculations are illustrated on a series of numerical experiments.
We consider a model explaining the dependence between the productivity of labor (PL) and the fixed capital per worker (FC) in an economic system. The core of the model is a nonlinear oscillator with a limit cycle as an attractor. We run numerical simulations of the dynamics specific to this non-autonomous model to compare with the actual data recorded for the years 1987–2001 for the enterprise Omsk Bacon. Based on the numerical analysis we can conclude that the interior dynamics is not affected by exterior perturbations. The numerical simulations can help the managers of the enterprise to take the right steps to avoid stagnation.
We consider the numerical approximation of linear quadratic optimal control problems for partial differential equations where the dynamics is driven by a strongly continuous semigroup. For this problems, the optimal control is given in feedback form, i.e., it relies on solving the associated Riccati equation and the optimal state. We propose innovative integrators for solving the optimal state based on operator splitting procedures and exponential integrators and prove their convergence. We illustrate the performance of our approach in numerical experiments.
There is studied the inverse problem to determine solution {T(z,t), μ (t)} of the equation ρ (z) C(z) T_t = (k(z) T_z)_z - ρ (z) C(z) w(z) T_z, (t,z) ∈ Q ≡ [0,t_f] × [0,H], with initial condition T(z,0)=U(z), z ∈ [0,H], and boundary conditions T(0,t)=U_s+μ (t), -k(H) T_z(H,t)=q, t ∈ [0,t_f], and redetermination condition T(z,t_f)=χ (z), z ∈ [0,H], where w(z) is the advection rate of glacier layers, ρ (z) , C(z), and k(z) are the density, specific heat, and thermal conductivity of ice, respectively, q is the geothermal heat flux, U_s is the steady-state surface temperature in the past, t_f is the present time, H is the borehole depth, χ (z) is the measured temperature-depth profile. The solution of the inverse problem μ (t) is looking for in the finite Fourier series form where periods correspond to the climatic signals retrieved by the annual tree-ring width index. It is derived that the solution is unique and stable and can be found out by the Tikhonov’s regularization method. This method is applied for the Kamchatka region.
Slot machines are casino gambling machines with three or more reels which spin when a button is pushed. The machine pays are based on patterns of symbols visible on the front of the machine when it stops. Most of the modern slots consist of a base game, free games and bonus games. The base game is the core of the playing process. Player’s money are usually taken as bet in the base game and no bet is taken during free games or bonus games. Each slot machine has a parameter called return to player (RTP). RTP is the average amount of money which a player will get back, in average, after each spin of the reels. The total RTP (measured in percents) can be in the range between 75% and 98%. Its components are: base game RTP, free games RTP, bonus games RTP. The base game also controls how often free games will be activated and how often bonus games will be played. In this paper an evolutionary optimization algorithm for optimization of slot machine base game RTP by rearrangement of the symbols in the reels, is proposed. The problem itself is a combinatorial problem and the fitness function used checks all possible slot machine winning screens.
