In this paper, we propose a high-order finite volume method for solving multicomponent fluid problems. Our method couples the quasi-conservative form with the reconstruction of conservative variables in a characterist...
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In this paper, we propose a high-order finite volume method for solving multicomponent fluid problems. Our method couples the quasi-conservative form with the reconstruction of conservative variables in a characteristic manner. The source term and numerical fluxes are carefully designed to maintain the pressure and velocity equilibrium for the interface-only problem and preserve the equilibrium of physical parameters in a single-component fluid. These ingredients enable our scheme to achieve both high-order accuracy in the smooth region and the high resolution in the discontinuity region of the solution. Extensive numerical tests are performed to verify the high resolution and accuracy of the scheme.
In this paper, we design and analyze a space-time spectral method for the subdiffusion ***, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. ...
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In this paper, we design and analyze a space-time spectral method for the subdiffusion ***, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time-consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable ψ-fractional Sobolev spaces and a new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows us to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials(GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity.
This work focuses on the temporal average of the backward Euler-Maruyama(BEM)method,which is used to approximate the ergodic limit of stochastic ordinary differential equations(SODEs).We give the central limit theorem...
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This work focuses on the temporal average of the backward Euler-Maruyama(BEM)method,which is used to approximate the ergodic limit of stochastic ordinary differential equations(SODEs).We give the central limit theorem(CLT)of the temporal average of the BEM method,which characterizes its asymptotics in *** the deviation order is smaller than the optimal strong order,we directly derive the CLT of the temporal average through that of original equations and the uniform strong order of the BEM *** the case that the deviation order equals to the optimal strong order,the CLT is established via the Poisson equation associated with the generator of original *** experiments are performed to illustrate the theoretical *** main contribution of this work is to generalize the existing CLT of the temporal average of numerical methods to that for SODEs with super-linearly growing drift coefficients.
In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation *** can maintain the high accuracy in t...
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In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation *** can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears,and it is compact which will be good for giving the numerical boundary ***,it avoids complicated least square procedure when we implement the genuine two dimensional(2D)finite volume HWENO reconstruction,and it can be regarded as a generalization of the one dimensional(1D)HWENO *** numerical tests are performed to verify the high resolution and high accuracy of the scheme.
In this paper the author investigates the following predator-prey model with prey-taxis and rotational?ux terms■in a bounded domain with smooth *** presents the global existence of generalized solutions to the model...
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In this paper the author investigates the following predator-prey model with prey-taxis and rotational?ux terms■in a bounded domain with smooth *** presents the global existence of generalized solutions to the model■in any dimension.
The speeding-up and slowing-down(SUSD)direction is a novel direction,which is proved to converge to the gradient descent direction under some *** authors propose the derivative-free optimization algorithm SUSD-TR,whic...
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The speeding-up and slowing-down(SUSD)direction is a novel direction,which is proved to converge to the gradient descent direction under some *** authors propose the derivative-free optimization algorithm SUSD-TR,which combines the SUSD direction based on the covariance matrix of interpolation points and the solution of the trust-region subproblem of the interpolation model function at the current iteration *** analyze the optimization dynamics and convergence of the algorithm *** of the trial step and structure step are *** results show their algorithm’s efficiency,and the comparison indicates that SUSD-TR greatly improves the method’s performance based on the method that only goes along the SUSD *** algorithm is competitive with state-of-the-art mathematical derivative-free optimization algorithms.
Gradient method is an important method for solving large scale problems. In this paper, a new gradient method framework for unconstrained optimization problem is proposed, where the stepsize is updated in a cyclic way...
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In this paper,the authors consider the stabilization and blow up of the wave equation with infinite memory,logarithmic nonlinearity and acoustic boundary *** authors discuss the existence of global solutions for the i...
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In this paper,the authors consider the stabilization and blow up of the wave equation with infinite memory,logarithmic nonlinearity and acoustic boundary *** authors discuss the existence of global solutions for the initial energy less than the depth of the potential well and investigate the energy decay estimates by introducing a Lyapunov ***,the authors establish the finite time blow up results of solutions and give the blow up time with upper bounded initial energy.
Thermal phase change problems are widespread in mathematics,nature,and *** are particularly useful in simulating the phenomena of melting and solidification in materials *** this paper we propose a novel class of arbi...
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Thermal phase change problems are widespread in mathematics,nature,and *** are particularly useful in simulating the phenomena of melting and solidification in materials *** this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase changemodel,which is the coupling of a heat transfer equation and a phase field *** unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes.A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step,with an efficient scheme of sufficient accuracy to calculate the solution at the first *** is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time *** time step size strategies can be applied to further benefit from this unconditional *** experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.
In this paper,we consider numerical solutions of the fractional diffusion equation with theαorder time fractional derivative defined in the Caputo-Hadamard sense.A high order time-stepping scheme is constructed,analy...
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In this paper,we consider numerical solutions of the fractional diffusion equation with theαorder time fractional derivative defined in the Caputo-Hadamard sense.A high order time-stepping scheme is constructed,analyzed,and numerically *** contribution of the paper is twofold:1)regularity of the solution to the underlying equation is investigated,2)a rigorous stability and convergence analysis for the proposed scheme is performed,which shows that the proposed scheme is 3+αorder *** numerical examples are provided to verify the theoretical statement.
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