Local Truncation Error Analysis Research Articles

Hexagonal finite differences for the two-dimensional variable coefficient Poisson equation

For many years, finite differences in hexagonal grids have been developed to solve elliptic problems such as the Poisson and Helmholtz equations. However, these schemes are limited to constant coefficients, which reduces their usefulness in many applications. The main challenge is accurately approximating the diffusive term. This paper presents examples of both successful and unsuccessful attempts to obtain accurate finite differences based on a hexagonal stencil with equilateral triangles to approximate two-dimensional Poisson equations. Local truncation error analysis reveals that a second-order scheme can be achieved if the derivative of the diffusive coefficient is included. Finally, we provide numerical examples to verify the accuracy of the proposed methods.

Numerical Solution for the Heat Equation Using Crank-Nicolson Difference Method

Abstract: In this paper, the Crank-Nicolson difference method is applied to a simple problem involving one dimensional heat equation. Numerical solution to the heat equation using Crank-Nicolson difference equation is obtained. The method of solving the problem is implemented by using Python Programming. We have discussed the local truncation error and stability analysis of Crank-Nicolson difference method difference method of heat equation.

CFL Optimized Forward–Backward Runge–Kutta Schemes for the Shallow-Water Equations

Abstract We present the formulation and optimization of a Runge–Kutta-type time-stepping scheme for solving the shallow-water equations, aimed at substantially increasing the effective allowable time step over that of comparable methods. This scheme, called FB-RK(3,2), uses weighted forward–backward averaging of thickness data to advance the momentum equation. The weights for this averaging are chosen with an optimization process that employs a von Neumann–type analysis, ensuring that the weights maximize the admittable Courant number. Through a simplified local truncation error analysis and numerical experiments, we show that the method is at least second-order in time for any choice of weights and exhibits low dispersion and dissipation errors for well-resolved waves. Further, we show that an optimized FB-RK(3,2) can take time steps up to 2.8 times as large as a popular three-stage, third-order strong stability-preserving Runge–Kutta method in a quasi-linear test case. In fully nonlinear shallow-water test cases relevant to oceanic and atmospheric flows, FB-RK(3,2) outperforms SSPRK3 in admittable time step by factors roughly between 1.6 and 2.2, making the scheme approximately twice as computationally efficient with little to no effect on solution quality. Significance Statement The purpose of this work is to develop and optimize time-stepping schemes for models relevant to oceanic and atmospheric flows. Specifically, for the shallow-water equations we optimize for schemes that can take time steps as large as possible while retaining solution quality. We find that our optimized schemes can take time steps between 1.6 and 2.2 times larger than schemes that cost the same number of floating point operations, translating directly to a corresponding speedup. Our ultimate goal is to use these schemes in climate-scale simulations.

Numerical Solution for the Heat Equation Using Forward Time Centered Space Difference Method

Abstract: In this paper, the forward time centered space is applied to a simple problem involving one dimensional heat equation. Numerical solution to the heat equation using forward time centered space difference equation is obtained. The method of solving the problem is implemented by using Python Programming. We have discussed the local truncation error and stability analysis of explicit forward time centered space difference method of heat equation.

A superconvergent finite node method for semilinear elliptic problems

This paper proposes and analyzes a superconvergent finite node method (SFPM) for meshless solution of semilinear boundary value problems with variable coefficients. Smoothed derivatives of the moving least squares approximation are adopted in the SFPM to achieve the high accuracy and favorable superconvergence. Theoretical accuracy analysis of the SFPM and the traditional FPM is detailedly presented through local truncation error analysis. Some numerical results are provided to demonstrate the superconvergence and theoretical results of the method.

An accuracy analysis framework for meshfree collocation methods with particular emphasis on boundary effects

A theoretical framework for the accuracy analysis of meshfree collocation methods is presented with a particular focus on the effects of boundary conditions. In the proposed framework, the error of a meshfree collocation formulation is decomposed into three parts, namely, the collocation errors associated with the interior nodes, the Dirichlet boundary nodes and the Neumann boundary nodes, respectively. The errors arising from different parts of the computational domain can be conveniently obtained by performing a local truncation error analysis on the discrete meshfree collocation equations corresponding to the generic interior and boundary nodes. It turns out that the accuracy of a meshfree collocation formulation is controlled by the lowest accuracy order resulting from the collocation errors at both interior and boundary nodes, as well as the meshfree interpolation errors. Along this path, it is shown that accuracy of the standard node-based meshfree collocation method is governed by the collocation error related to interior nodes for both L2 and H1 error norms. Nonetheless, for different boundary conditions, the accuracy of the subdomain meshfree collocation method may rely on either the collocation error associated with interior nodes, the collocation errors corresponding to Dirichlet and Neumann boundary nodes, or the interpolation errors. It is further proved that the optimal convergence rates of Galerkin meshfree formulation can be achieved by the subdomain meshfree collocation method if the problems are purely subjected to Dirichlet boundary conditions. The proposed theoretical results are well confirmed by numerical examples.

A High-Order Approximate Solution for the Nonlinear 3D Volterra Integral Equations with Uniform Accuracy

In this paper, we present a high-order approximate solution with uniform accuracy for nonlinear 3D Volterra integral equations. This numerical scheme is constructed based on the three-dimensional block cubic Lagrangian interpolation method. At the same time, we give the local truncation error analysis of the numerical scheme based on Taylor’s theorem. Through theoretical analysis, we reach the conclusion that the optimal convergence order of this high-order numerical scheme is 4. Finally, we verify the effectiveness and applicability of the method through four numerical examples.

Physics-based sizing functions for ocean-to-estuary shallow-water models

Modeling ocean processes requires a discrete representation of the system geometry, which itself contains multiple degrees of freedom, especially for systems of large geographical scale. This paper assesses the unstructured meshing requirements for a large-scale, astronomic-tide model domain of the western North Atlantic Ocean, Gulf of Mexico and Caribbean Sea with telescopic focus of the South Atlantic Bight (United States southeastern seaboard) and its estuaries. The methodologies of localized truncation error analysis (LTEA), which account only for the model interior and LTEA plus complex derivatives (+CD), which account for the model interior and boundary, are exploited due to their foundations stemming directly from tidal physics. LTEA and LTEA+CD are applied for the M2 tidal harmonics resulting from linear and nonlinear solutions of the ADCIRC (advanced circulation) model based on a highly resolved uniform mesh. The resultant target element size distributions are interrogated distinctively for the inshore (estuarine) region of the South Atlantic Bight, the inlet-punctuated coastline, the shelf-slope-rise triad and deep ocean, and the open-ocean boundary. The relatively large size of the LTEA computational molecule makes it the suitable choice for the deep ocean, offshore and shelf regions where the tidal dynamics are predominantly linear and smoothly changing in space. LTEA+CD is the logical choice for the shallows and estuaries because of the spatial coverage afforded by the combined utility of interior and boundary estimators (computational molecules), as well as its accounting for the nonlinearities in the evaluation of local truncation error. The combination of the two methods enables an almost complete coverage of the domain for generating spacing requirements, with LTEA+CD extending beyond the interior-only definition of LTEA to capture the inshore regions and model boundaries. Implementing the approach with LTEA and LTEA+CD to target finer resolution in areas with sharper gradients in the hydrodynamics, while relaxing resolution in areas with smoother hydrodynamics, resulted in a mesh with a far-reduced number of elements (∼1.3M) relative to an overly resolved uniform mesh (>10M elements) that is comparable in accuracy with ±5% tidal amplitude error for 99% of the domain.

An efficient variable stepsize rational method for stiff, singular and singularly perturbed problems

In this article, a new iterative method of the rational type having fifth-order of accuracy is proposed to solve initial value problems. The method is self-starting, stable, consistent, and convergent, whereas local truncation error analysis has also been discussed. Furthermore, the method has been analyzed with a variable stepsize approach that increases performance while taking fewer steps with acceptable local errors. The method is also tested against some existing fifth-order methods having rational structure. The proposed one outperforms concerning maximum absolute error, final absolute error, average error, and norm, while CPU time computed in seconds is comparable. Furthermore, stiff, singular, and singularly perturbed problems for single and system of differential equations chosen for simulations yielded minor errors when solved with the new rational method.

Traveling wave solutions and numerical solutions of Gilson–Pickering equation

In this article, we introduce traveling wave solutions of the Gilson–Pickering equation using two simple methods. Furthermore, we present a numerical study of the Gilson–Pickering equation using the finite difference method. Also, the local truncation error and stability analysis for the difference scheme are presented. We justify the accuracy of the obtained results by exhibiting illustrative tables and proper graphs.

A Class of Spline Functions for Solving 2-Order Linear Differential Equations with Boundary Conditions

In this paper, we exploit an numerical method for solving second order differential equations with boundary conditions. Based on the theory of the analytic solution, a series of spline functions are presented to find approximate solutions, and one of them is selected to approximate the solution automatically. Compared with the other methods, we only need to solve a tri-diagonal system, which is much easier to implement. This method has the advantages of high precision and less computational cost. The analysis of local truncation error is also discussed in this paper. At the end, some numerical examples are given to illustrate the effectiveness of the proposed method.

A high-order compact alternating direction implicit method for solving the 3D time-fractional diffusion equation with the Caputo–Fabrizio operator

In this paper, a high-order compact finite difference method (CFDM) with an operator-splitting technique for solving the 3D time-fractional diffusion equation is considered. The Caputo–Fabrizio time operator is evaluated by the $$L_1$$ approximation, and the second-order space derivatives are approximated by the compact CFDM to obtain a discrete scheme. Alternating direction implicit method (ADI) is used to split the problem into three separate one-dimensional problems. The local truncation error analysis is discussed. Moreover, the convergence and stability of the numerical method are investigated. Finally, some numerical examples are presented to demonstrate the accuracy of the compact ADI method.

A new multistep method with optimized characteristics for initial and/or boundary value problems

In this paper we introduce, for the first time in the literature, an optimized multistage symmetric two-step method. This method is considered as optimized due to the following reasons: (1) it is of tenth-algebraic order scheme, (2) it has obliterated the phase-lag and its first, second, third and fourth derivatives, (3) it has improved stability characteristics, (4) it is a P-stable method. For the new proposed multistage symmetric two-step method we present a full theoretical investigation consisted of: (1) local truncation error and comparative error analysis, (2) stability analysis and (3) interval of periodicity analysis. The effectiveness of the new builded multistage symmetric two-step method is evaluated on the solution of systems of coupled differential equations of the Schrodinger type.

New Runge–Kutta type symmetric two step finite difference pair with improved properties for second order initial and/or boundary value problems

A new three-stages symmetric two-step method with improved properties is developed in this paper and for the first time in the literature. The properties of the new proposed algorithm are: The new proposed scheme is constructed based on the following layers: For the new proposed method we give a full theoretical analysis which consists of: (1) local truncation error analysis, (2) comparative local truncation error analysis, (3) stability analysis and (4) interval of periodicity analysis. The efficiency of the new proposed algorithm is tested on the approximate solution of systems of coupled differential equations arising from the Schrodinger equation.

Efficient Krylov-based exponential time differencing method in application to 3D advection-diffusion-reaction systems

The number of ordinary differential equations generally increases exponentially as the partial differential equation is posed on a domain with more dimensions. This is, of course, the curse of dimensionality for exponential time differencing methods. The computational challenge in applying exponential time differencing methods for solving partial differential equations in high spatial dimensions is how to compute the matrix exponential functions for very large matrices accurately and efficiently. In this paper, our main aim is to design a Krylov subspace approximation-based locally extrapolated exponential time differencing method and compare its performance in terms of accuracy and efficiency to the already available method in the literature for solving a three-dimensional nonlinear advection-diffusion-reaction systems. The fundamental idea of the proposed method is to compute only the action of the matrix exponential on a given state vector instead of computing the matrix exponential itself, and then multiplying it with given vector. The stability and local truncation error of the proposed method have been examined. Calculation of local truncation error and empirical convergence analysis indicate the proposed method is second-order accurate in time. The performance and reliability of this novel method have been investigated by testing it on systems of the three-dimensional nonlinear advection-diffusion-reaction equations and three-dimensional viscous nonlinear Burgers’ equation.

Superconvergent gradient smoothing meshfree collocation method

A superconvergent meshfree collocation method with smoothed nodal gradients is presented. In this method, the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function, while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients. It is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function, which facilitates a trivial numerical implementation. Subsequently, an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points. Based upon a local truncation error analysis, it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions. A key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation. Another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation, which is non-feasible in the conventional collocation formulation. The effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems. Numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method.

New hybrid symmetric two step scheme with optimized characteristics for second order problems

In the present paper, for the first time in the literature, we build a new three-stages symmetric two-step finite difference pair with optimized properties. In more details the new method: (1) is of symmetric type, (2) is of two-step algorithm, (3) is of three-stages—i.e. hybrid or Runge–Kutta type, (4) it is of tenth-algebraic order, (5) it has vanished the phase-lag and its first and second derivatives, (6) it has optimized stability properties for the general problems, (7) it is a P-stable finite difference scheme since it has an interval of periodicity equal to $$\left( 0, \infty \right) $$ . The new Runge–Kutta type algorithm is builded based on the following approximations: A full theoretical analysis (local truncation error analysis, comparative error analysis and stability and interval of periodicity analysis) is given for the new builded finite difference pair. The effectiveness of the new builded hybrid scheme is evaluated on the numerical solution of systems of coupled differential equations of the Schrodinger type.

New three–stages symmetric two step method with improved properties for second order initial/boundary value problems

In this paper and for the first time in the literature we develop a new three–stages symmetric two–step method with improved properties. More specifically the new scheme: (1) is of symmetric type, (2) is of two–step algorithm, (3) is of three–stages, (4) it is of tenth–algebraic order, (5) it has eliminated the phase–lag and its first derivative, (6) it has improved stability properties for the general problems, (7) it is a P–stable method since it has an interval of periodicity equal to $$\left( 0, \infty \right) $$ . In order to develop the new hybrid algorithm we use the following approximations: (i) An approximation developed on the first layer on the point $$x_{n-1}$$ , (ii) An approximation developed on the second layer on the point $$x_{n}$$ and finally, (iii) An approximation developed on the third (final) layer on the point $$x_{n+1}$$ . For the new proposed method we give a full theoretical analysis (local truncation error analysis and stability and interval of periodicity analysis). The efficiency of the new proposed method is examined on the numerical solution of coupled differential equations arising from the Schrodinger equation.

Automatic, unstructured mesh optimization for simulation and assessment of tide- and surge-driven hydrodynamics in a longitudinal estuary: St. Johns River

A localized truncation error analysis with complex derivatives (LTEA+CD) is applied recursively with advanced circulation (ADCIRC) simulations of tides and storm surge for finite element mesh optimization. Mesh optimization is demonstrated with two iterations of LTEA+CD for tidal simulation in the lower 200 km of the St. Johns River, located in northeast Florida, and achieves more than an over 50% decrease in the number of mesh nodes, relating to a twofold increase in efficiency, at a zero cost to model accuracy. The recursively generated meshes using LTEA+CD lead to successive reductions in the global cumulative truncation error associated with the model mesh. Tides are simulated with root mean square error (RMSE) of 0.09–0.21 m and index of agreement (IA) values generally in the 80s and 90s percentage ranges. Tidal currents are simulated with RMSE of 0.09–0.23 m s−1 and IA values of 97% and greater. Storm tide due to Hurricane Matthew 2016 is simulated with RMSE of 0.09–0.33 m and IA values of 75–96%. Analysis of the LTEA+CD results shows the M2 constituent to dominate the node spacing requirement in the St. Johns River, with the M4 and M6 overtides and the STEADY constituent contributing some. Friction is the predominant physical factor influencing the target element size distribution, especially along the main river stem, while frequency (inertia) and Coriolis (rotation) are supplementary contributing factors. The combination of interior- and boundary-type computational molecules, providing near-full coverage of the model domain, renders LTEA+CD an attractive mesh generation/optimization tool for complex coastal and estuarine domains. The mesh optimization procedure using LTEA+CD is automatic and extensible to other finite element-based numerical models. Discussion is provided on the scope of LTEA+CD, the starting point (mesh) of the procedure, the user-specified scaling of the LTEA+CD results, and the iteration (termination) of LTEA+CD for mesh optimization.

New hybrid two-step method with optimized phase and stability characteristics

In this paper and for the first time in the literature, we develop a new Runge–Kutta type symmetric two-step finite difference pair with the following characteristics: For the new developed algorithm we present a detailed numerical analysis (local truncation error and stability analysis). The effectiveness of the new developed algorithm is evaluated with the approximate solution of coupled differential equations arising from the Schrodinger type.