scholarly journals On the Computation of the Mountain Torque from Gridded Global Datasets

2008 ◽  
Vol 136 (10) ◽  
pp. 4005-4009 ◽  
Author(s):  
Huei-Ping Huang ◽  
Klaus M. Weickmann
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Spectral Method ◽  
Surface Pressure ◽  
Finite Difference Scheme ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Axial Component ◽  
Practical Applications ◽  
Benchmark Calculation

This note evaluates the numerical schemes used for computing the axial component of the mountain torque from gridded global surface pressure and topography datasets. It is shown that the two formulas of the mountain torque based on (i) an integral of the product of the surface pressure and the gradient of topography, and (ii) an integral of the product of the topography and the surface pressure gradient, should produce identical results if a centered even-ordered finite-difference scheme or the spectral method is used to evaluate the integrand. Noncentered finite-difference schemes are not recommended not only because they produce extremely large errors but also because they produce different results for the two formulas. When compared with the benchmark calculation using the spectral method, it is found that the centered fourth-order finite-difference scheme is an efficient and generally accurate approximation for practical applications. Using the data from NCEP–NCAR reanalysis, the finite-difference schemes generally underestimate the global mountain torque compared to the benchmark. This negative error is interpreted as due to the asymmetry in the distribution of surface pressure and in the steepness of the topography between the western and eastern slopes of the mountains.

COMPARISON OF FINITE‐DIFFERENCE SCHEMES FOR THE GROSS‐PITAEVSKII EQUATION

2009 ◽  
Vol 14 (1) ◽  
pp. 109-126 ◽  
Author(s):  
Vyacheslav A. Trofimov ◽  
Nikolai Peskov
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Conservation Laws ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Pitaevskii Equation ◽  
Conservative Scheme ◽  
Conservative Finite Difference Scheme

A conservative finite‐difference scheme for numerical solution of the Gross‐Pitaevskii equation is proposed. The scheme preserves three invariants of the problem: the L 2 norm of the solution, the impulse functional, and the energy functional. The advantages of the scheme are demonstrated via several numerical examples in comparison with some other well‐known and widely used methods. The paper is organized as follows. In Section 2 we consider three main conservation laws of GPE and derive the evolution equations for first and second moments of a solution of GPE. In Section 3 we define the conservative finite‐difference scheme and prove the discrete analogs of conservation laws. The remainder of Section 3 consists of a brief description of other finite‐difference schemes, which will be compared with the conservative scheme. Section 4 presents the results of numerical solutions of three typical problems related to GPE, obtained by different methods. Comparison of the results confirms the advantages of conservative scheme. And finally we summarize our conclusions in Section 5.

Two Finite Difference Schemes for Multi-Dimensional Fractional Wave Equations with Weakly Singular Solutions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jinye Shen ◽  
Martin Stynes ◽  
Zhi-Zhong Sun
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Mesh Size ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Initial Time ◽  
Finite Difference Schemes

Abstract A time-fractional initial-boundary value problem of wave type is considered, where the spatial domain is ( 0 , 1 ) d (0,1)^{d} for some d ∈ { 1 , 2 , 3 } d\in\{1,2,3\} . Regularity of the solution 𝑢 is discussed in detail. Typical solutions have a weak singularity at the initial time t = 0 t=0 : while 𝑢 and u t u_{t} are continuous at t = 0 t=0 , the second-order derivative u t ⁢ t u_{tt} blows up at t = 0 t=0 . To solve the problem numerically, a finite difference scheme is used on a mesh that is graded in time and uniform in space with the same mesh size ℎ in each coordinate direction. This scheme is generated through order reduction: one rewrites the differential equation as a system of two equations using the new variable v := u t v:=u_{t} ; then one uses a modified L1 scheme of Crank–Nicolson type for the driving equation. A fast variant of this finite difference scheme is also considered, using a sum-of-exponentials (SOE) approximation for the kernel function in the Caputo derivative. The stability and convergence of both difference schemes are analysed in detail. At each time level, the system of linear equations generated by the difference schemes is solved by a fast Poisson solver, thereby taking advantage of the fast difference scheme. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of both numerical methods.

scholarly journals ON CONSTRUCTION AND ANALYSIS OF FINITE DIFFERENCE SCHEMES FOR PSEUDOPARABOLIC PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

2014 ◽  
Vol 19 (2) ◽  
pp. 281-297 ◽  
Author(s):  
Raimondas Čiegis ◽  
Natalija Tumanova
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Two Dimensional ◽  
The Stability

In this paper the one- and two-dimensional pseudoparabolic equations with nonlocal boundary conditions are approximated by the Euler finite difference scheme. In the case of classical boundary conditions the stability of all schemes is investigated by the spectral method. Stability regions of finite difference schemes approximating pseudoparabolic problem are compared with the stability regions of the classical discrete parabolic problem. These results are generalized for problems with nonlocal boundary conditions if a matrix of the finite difference scheme can be diagonalized. For the two-dimensional problem an efficient algorithm is constructed, which is based on the combination of the FFT method and the factorization algorithm. General stability results, known for the three level finite difference schemes, are applied to investigate the stability of some explicit approximations of the two-dimensional pseudoparabolic problem with classical boundary conditions. A connection between the energy method stability conditions and the spectrum Hurwitz stability criterion is shown. The obtained results can be applied for pseudoparabolic problems with nonlocal boundary conditions, if a matrix of the finite difference scheme can be diagonalized.

scholarly journals Exact Finite Difference Scheme and Nonstandard Finite Difference Scheme for Burgers and Burgers-Fisher Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Lei Zhang ◽  
Lisha Wang ◽  
Xiaohua Ding
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Burgers Equation ◽  
Solitary Wave Solution ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Fisher Equation ◽  
Nonstandard Finite Difference Scheme ◽  
Nonstandard Finite Difference

We present finite difference schemes for Burgers equation and Burgers-Fisher equation. A new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation is proposed using the solitary wave solution. Then nonstandard finite difference schemes are constructed to solve two equations. Numerical experiments are presented to verify the accuracy and efficiency of such NSFD schemes.

The Finite Difference Scheme for 3d Mathematical Modeling of a Wood Drying Process

2001 ◽  
Vol 1 (2) ◽  
pp. 125-137 ◽  
Author(s):  
Raimondas Čiegis ◽  
Vadimas Starikovičius
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Experiments ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Drying Process ◽  
The Stability ◽  
Robin Boundary

AbstractThis work discusses issues on the design and analysis of finite difference schemes for 3D modeling the process of moisture motion in the wood. A new finite difference scheme is proposed. The stability and convergence in the maximum norm are proved for Robin boundary conditions. The influence of boundary conditions is investigated, and results of numerical experiments are presented.

On the Pseudo-Spectral Method and Spectral Accuracy

Geophysics ◽  
2021 ◽  
pp. 1-2
Author(s):  
Rune Mittet
Keyword(s):  
Finite Difference ◽  
Spectral Method ◽  
Material Parameter ◽  
Step Function ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Spectral Accuracy ◽  
Heaviside Step Function ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral

There are numerical accuracy problems related to the implementation of sharp internal interfaces in pseudo-spectral and finite-difference schemes. It is common practice to classify numerical errors due to the implementation of interfaces as being to some order in a Taylor expansion. An alternative approach is to classify these errors as being to some order in a Fourier expansion.The pseudo-spectral method does not provide spectral accuracy in inhomogeneous media. The numerical errorsfor the upper half of the frequency/wavenumber spectra of the propagating fields are not related to theimplementation of the derivative operators but to aliasing effects coming from the multiplicationof static material-parameter fields with the dynamic, propagating, fields. The pseudo-spectral methodcan only provide half-spectral accuracy. The same type of spatial aliasing errors are present also forfinite-difference schemes. High-order finite differences can provide the same accuracy as the pseudo-spectral method if the staggered finite-difference derivative operators have a negligible errorat four grid points per shortest wavelength and above. Smoothing of the material-parameter field leads to additional reduction in the error-free bandwidthof the propagating fields. Assuming that there is a maximum wavenumber up to which the spectrumof the smoothed model coincide with the implementation using a properly bandlimited Heaviside step function, then there exists a local critical wavenumber for the propagating field equal toone half of the maximum wavenumber for the smoothed model. Harmonic averaging of material-parameter fields also results in wavenumber spectra where there is a maximum wavenumber above whichthe wavenumber spectrum deviates from an implementation with a bandlimited Heaviside step function.The same one-half rule is applicable also in this case.

THE FINITE DIFFERENCE SCHEME FOR MATHEMATICAL MODELING OF WOOD DRYING PROCESS

2001 ◽  
Vol 6 (1) ◽  
pp. 48-57 ◽  
Author(s):  
R. Čiegis ◽  
V. Starikovičius
Keyword(s):  
Mathematical Modeling ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Drying Process ◽  
Moisture Movement ◽  
Different Types ◽  
The Stability

This work discusses issues on the design of finite difference schemes for modeling the moisture movement process in the wood. A new finite difference scheme is proposed. The stability and convergence in the maximum norm are proved for different types of boundary conditions.

scholarly journals On the stability of explicit finite difference schemes for a pseudoparabolic equation with nonlocal conditions

2014 ◽  
Vol 19 (2) ◽  
pp. 225-240 ◽  
Author(s):  
Justina Jachimavičienė ◽  
Mifodijus Sapagovas ◽  
Artūras Štikonas ◽  
Olga Štikonienė
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Nonlinear Eigenvalue Problem ◽  
Nonlocal Conditions ◽  
Finite Difference Schemes ◽  
Pseudoparabolic Equation ◽  
Third Order ◽  
Explicit Finite Difference ◽  
The Stability

A new explicit conditionally consistent finite difference scheme for one-dimensional third-order linear pseudoparabolic equation with nonlocal conditions is constructed. The stability of the finite difference scheme is investigated by analysing a nonlinear eigenvalue problem. The stability conditions are stated and stability regions are described. Some numerical experiments are presented in order to validate theoretical results.

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