# حل عددی معادلات آب کم‌عمق با استفاده از روش فشرده ترکیبی مرتبه ششم

نوع مقاله : مقاله تحقیقی‌ (پژوهشی‌)

نویسندگان

1 موسسه ژئوفیزیک دانشگاه تهران، ایران

2 گروه فیزیک دانشگاه آزاد اسلامی واحد شوشتر، ایران

چکیده

در این تحقیق، حل عددی معادلات آب کم‌عمق غیرخطی در صفحه f برحسب میدان‌های ارتفاع، واگرایی و تاوایی با استفاده از روش فشرده ترکیبی مرتبه ششم مورد بررسی قرار می‌گیرد و نتایج آن با روش‌های مرتبه دوم مرکزی، فشرده مرتبه چهارم، اَبَرفشرده مرتبه ششم و طیفی‌وار مقایسه می‌شود. برای این منظور، یک جت مداری به‌منزلة شرایط اولیه درنظر گرفته می‌شود که با گذشت زمان به ساختارهایی پیچیده با مقیاس کوچک‌تر شکسته می‌شود. در این حل عددی، برای انتگرال‌گیری زمانی معادلات از فرمول‌بندی نیمه‌ضمنی سه‌ترازه استفاده شده‌ است. در مورد معادله تاوایی، یک جمله فراپخش برای حفظ پایداری به حل عددی افزوده می‌شود. نتایج به‌دست آمده نشان از توانایی زیاد روش فشرده ترکیبی مرتبه ششم در شبیه‌سازی میدان‌های جریان پیچیده دارد. با وجود اینکه روش طیفی‌وار نسبت‌ به سایر روش‌ها دقت بیشتری دارد، نزدیکی بسیار زیاد نتایج روش فشرده ترکیبی مرتبه ششم به نتایج روش طیفی‌وار امیدوار کننده است.

کلیدواژه‌ها

عنوان مقاله [English]

### Numerical solution of shallow water equations using the sixth-order combined compact method

نویسندگان [English]

• Hakim Golshahy 2
چکیده [English]

Usually, simplified models, such as shallow water model, are used to describe atmospheric and oceanic motions. The shallow water equations are widely applied in various oceanic and atmospheric extents. This model is applied to a fluid layer of constant density in which the horizontal scale of the flow is much greater than the layer depth. However, the dynamics of a two-dimensional shallow water model is less general than three-dimensional general circulation models but is preferred because of its greater mathematical and computational simplicity.
Taking intrinsic complexity of fluids, recently, numerical researches have been focused on highly accurate methods. Especially, for large grid spacing numerical simulation, the use of highly accurate methods have become urgent. This trend led to an interest in compact finite difference methods. The compact finite-difference schemes are simple and powerful ways to reach the objectives of high accuracy and low computational cost. Compared with the traditional explicit finite-difference schemes of the same-order, compact schemes have proved to be significantly more accurate along with the benefits of using smaller stencil sizes, which can be essential in treating non-periodic boundary conditions. Application of some families of the compact schemes to the spatial differencing in some idealized models of the atmosphere and oceans shows that compact finite difference schemes can be considered as a promising method for the numerical simulation of geophysical fluid dynamics problems.
In this research work, the sixth-order combined compact (CCD6) finite difference method was applied to the spatial differencing of f-plane shallow-water equations in vorticity, divergence and height forms (on a Randall's Z grid). The second-order centered (E2S), fourth-order compact (C4S) and sixth-order super compact (SCD6) finite difference methods were also used for spatial differencing of the shallow water equations and the results were compared to the ones from a pseudo-spectral (PS) method. A perturbed unstable zonal jet was considered as the initial condition for numerical simulation in which it breaks up into smaller vortices and becomes very complex. The shallow water equations are integrated in time using a three-level semi-implicit formulation. To control the build-up of small-scale activities and thus potential for numerical nonlinear instability, the non-dissipative vorticity equation was made dissipative by adding a hyperdiffusion term. The global distribution of mass between isolevels of the potential vorticity, called mass error, was used to assess numerical accuracy. The CCD6 generated the least mass error among finite difference methods used in this research. By taking the PS method as a reference, the qualitative and quantitative comparison of the results of the CCD6, SCD6, C4S and E2S, indicated the high accuracy of the sixth-order combined compact finite difference method.

کلیدواژه‌ها [English]

• Combined compact method
• numerical accuracy
• finite difference
• semi-implicit
• shallow water equations
• Z grid
• zonal jet

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