one dimensional quantum Ising model with nearest neighbor interaction in transverse magnetic field is one of the simplest spin models which undergo quantum phase transition. This model has been precisely solved using different methods. In this paper, we solve this model in uniform magnetic field -Jg sxn precisely using a new method called Continuous Unitary Transformations (CUT) or flow equations, and derive its expectation values <sxn>, <szn> and <sxnsxn+j>. Then, we apply this method on one dimensional quantum Ising model in staggered magnetic field (-1)nJgsxn. Results show that both models have the same critical properties as expected, and it was also found that spontaneous symmetry breaking cannot be derived from CUT.

In this paper, we develop a collocation method based on cubic B-spline to the solution of nonlinear parabolic equation "euxx=a(x, t)ut+F(x, t, u, ux) subject to appropriate initial, and Dirichlet boundary conditions, where" e>0 is a small constant. We developed a new two-level three-point scheme of order O(k2+h2). The convergence analysis of the method is proved. Numerical results are given to illustrate the efficiency of our method computationally.

‎This paper is devoted to proposing hybridized discontinuous Galerkin (HDG) approximations for solving a system of coupled Burgers equations (CBE) in a closed interval‎. ‎The noncomplete discretized HDG method is designed for a nonlinear weak form of one-dimensional $x-$variable such that numerical fluxes are defined properly‎, ‎stabilization parameters are applied‎, ‎and broken Sobolev approximation spaces are exploited in this scheme‎. ‎Having necessary conditions on the stabilization parameters‎, ‎it is proven in a theorem and corollary that the proposed method is stable with imposed homogeneous Dirichlet and/or periodic boundary conditions to CBE‎. ‎The desired HDG method is stated by using the Crank-Nicolson method for time-variable discretization and the Newton-Raphson method for solving nonlinear systems‎. ‎Numerical experiences show that the optimal rate of convergence is gained for approximate solutions and their first derivatives‎.

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On the basis of a reproducing kernel space, an iterative algorithm for solving the one-dimensional linear and nonlinear Schrödinger equations is presented. The analytical solution is shown in a series form in the reproducing kernel space and the approximate solution is constructed by truncating the series. The convergence of the approximate solution to the analytical solution is also proved. The method is examined for the single soliton  solution and interaction of two solitons. Numerical experiments show that the proposed method is of satisfactory accuracy and preserves the conservation laws of charge and energy. The numerical results are compared with both the analytical and numerical solutions of some earlier papers in the literature.

In this paper a multiquadric quasi-interpolation (MQQI) scheme for solving the system of 1-D coupled nonlinear Burger’ s equations (CNBE) is presented. The scheme utilizes the derivative of the quasi-interpolation for approximating the spatial derivative and the Taylor series expansion for temporal derivatives. Simulations are presented to demonstrate the efficiency and applicability of the scheme. Also, we have shown that our scheme is superior to some numerical schemes already done.

Introduction: The shallow water equations are a set of hyperbolic balance laws that describe the behavior of water flow in shallow regions such as rivers, lakes, and oceans. Solving hyperbolic balance laws poses significant challenges due to the presence of non-conservative terms, shocks and discontinuities. Analytical solutions are limited to simplified cases, so numerical methods are often employed to solve these equations. Numerical schemes addressing these balance laws must ensure the well-balanced property (Bermudez and Vázquez 1994), ensuring that discretized numerical fluxes must exactly balance by the approximated source terms. These types of numerical schemes utilize upwind/flux splitting techniques to handle wave propagation and discontinuities. Such well-balanced approaches work well for supercritical or subcritical regions but are known to struggle when Riemann problem includes both (LeFloch and Thanh 2011)- particularly in trans-critical flows and hydraulic jumps. To address this, various treatments, such as entropy fixes, shock fitting techniques, have been developed. Notably, Akbari and Pirzadeh (2022) (Akbari and Pirzadeh, 2022)introduced a set of shockwave fixes to cure the numerical slowly moving shock anomaly. Their approach is advantageous in accurately capturing the hydraulic jump. However, such scheme is only first-order accurate, as higher-order schemes progress, it becomes necessary to extend such technique to greater accuracy in high-resolution schemes.Methodology: A second order well balanced numerical scheme has been designed for the shallow water equations using a semi-discrete MUSCL reconstruction. The first step in the semi-discrete finite volume method is to discretize the governing equations in space. For the one-dimensional shallow water equations, this involves dividing the computational domain into a set of control volumes and approximating the integral form of the conservation equations over each control volume. By considering the fluxes at the control volume interfaces and accounting for the source terms, a system of ordinary differential equations (ODEs) can be obtained. To ensure accurate and stable solutions, a second-order finite volume approach is employed for spatial discretization. The proposed approach aims to exactly preserve all steady states of shallow water equations while maintaining the second order of accuracy. To achieve this, we extend a recently developed fully well-balanced scheme, called HLL-MSF, to higher-order of accuracy. To upgrade the first-order HLL-MSF scheme to second order while maintaining the same well-balanced property of the first order one, a MUSCL reconstruction approach with a suitable weighted technique is proposed. The weighted approach allows the numerical scheme to revert to the first order scheme with shockwave fixes at hydraulic jumps or at trans-critical points. Appropriate flux limiters are also introduced to ensure the well-balanced property of the numerical scheme in smooth steady state cases. The method's accuracy and stability are attributed to these carefully chosen flux limiters and weighted coefficients. The final step in the semi-discrete finite volume method involves time integration to advance the solution in time. In this paper, the third order explicit Runge-Kutta method is chosen as the time integration scheme. By combining the second-order finite volume spatial discretization and the third-order explicit Runge-Kutta time integration scheme, the proposed finite volume method ensures higher-order accuracy in both space and time.Results and Discussion: To verify the well-balanced property and the second order of accuracy of the proposed numerical scheme several numerical examples and benchmarks found in the literature including both steady and unsteady cases are presented. For numerical experiments that have analytical or reference solutions, numerical errors are calculated using L1 and L∞ norms. The first test case is devoted to the simulation of steady state at rest or the lake at rest situation. Numerical errors demonstrate that the proposed scheme is exactly well-balanced in this case. The second test case addresses a smooth steady state of trans-critical flow over a bump. The proposed second order scheme is confirmed to capture the smooth steady state precisely (Table 1). We also perform experiments on trans-critical flow with hydraulic jump to see how the proposed scheme behaves when the solution contains a shock discontinuity. Unlike the traditional higher-order schemes which often use the pre-balanced shallow water formulation to achieve the exact conservation property on steady state cases at rest, the proposed second order scheme can capture both smooth and non-smooth (Hydraulic jump) parts exactly with no smears and oscillations (Table 1). An additional test case is conducted to confirm the second order accuracy of the numerical scheme. Table 2 Illustrates that the intended accuracy is clearly achieved. Finally, three numerical experiments are conducted in quasi-steady and unsteady conditions including slowly moving shocks over flat or discontinuous topography. The higher-order approximate solvers are known to achieve better accuracy for such flows than the first order counterparts.Conclusion: In this paper, second-order well-balanced numerical schemes are developed for the solution of one-dimensional shallow water equations. The approach accurately models different regimes of the flow accurately. The advantage of the proposed scheme over existing higher-order schemes is the fully well-balanced and entropy satisfying properties, where all steady states solutions are exactly preserved.

Flow Simulation and discontinuous shock capturing are important in shallow water equations. Common numerical schemes such as finite difference Preissmann scheme, without performing some modifications, cannot simulate discontinuities. Finite volume methods using Riemann solvers by taking advantage of the characteristics of solving the smooth areas as well, have the ability to simulate discontinuities. In this paper, the second order Roe model of Riemann solver was employed by applying the limiting functions to eliminate the spurious oscillations of the numerical simulation in the surface and subsurface flows (Saint-Venant equations in surface flow and Kostiakov-Lewis in subsurface flow). A Fortran code was developed for Roe-TVD method, the presented model was evaluated using the Preissmann scheme (an implicit finite difference scheme) and two sets of field data (Printz-323 and Walker) based on Root Mean Square Error (RMSE), Standard Error (SE) and Determination Coefficient (R² ). It was concluded that Roe model showed better results comparing to the Preissmann scheme in all of the simulations, particularly in outgoing runoff, RMES was improved up to 62%. The applied model was an explicit method and reduced running time and had the ability of application under different field conditions.

In this paper, a two-dimensional multi-wavelet is constructed in terms of Chebyshev polynomials. The constructed multi-wavelet is an orthonormal basis for L2 [0,1]2 space. By discretizing two-dimensional Fredholm integral equation reduce to a algebraic system. The obtained system is solved by the Galerkin method in the subspace of L2 [0, 1]2 by using twodimensional multi-wavelet bases. Because the bases of subspaces are orthonormal, so the above mentioned system has a small dimension and also high accuracy in approximating solution of integral equations. For one-dimensional case, a similar works are done in [4,5], which they have small dimension and high accuracy. In this article, we extend one-dimensional case to twodimensional by extending and by choosing good functions on two axes. Numerical results show that the above mentioned method has a good accuracy.