ارائه یک روش عددی با دقت مرتبه دو کاملا متعادل و سازگار با شرایط آنتروپی برای دستگاه یک بعدی آب کم عمق

در این پژوهش، یک روش عددی با دقت مرتبه دو برای حل معادله ‌های یک بعدی آب کم عمق با وجود جملات چشمه ناشی از تغییرات بستر توسعه داده شده است. برای حل عددی این معادله‌ها حفظ وضعیت های دائمی جریان بدور از نوسانات غیرفیزیکی اهمیت اساسی دارد چرا که بیشتر شبیه‌سازی های کاربردی از معادله‌های آب کم عمق شامل اغتشاش های جزئی از این وضعیت های دائمی هستند. در سال‌های اخیر تکنیک های متعدی برای حفظ جریان های دائمی پیشنهاد شده است. با این حال اکثر این روش‌ها قادر به حفظ تمام وضعیت دائمی نیستند و روش هایی هم که چنین ویژگی دارند حداکثر از مرتبه یک دقت می باشند. در این مقاله یک روش مرتبه دو که قادر به حفظ تمام وضعیت های دائمی است توسعه داده شد. آزمایش های متنوعی برای صحت سنجی روش پیشنهادی در شبیه سازی جریان های دائمی انجام شد. نتایج عددی نشان داد که روش عددی پیشنهادی قادر به حفظ تمام وضعیت های دائمی جریان است. روش مرتبه دو پیشنهادی همچنین از دقت بهتری به نسبت حل‌گرهای تخمینی مرتبه یک برای شبیه سازی جریان های شبه دائمی و غیر دائمی برخوردار می‌باشد.

a second order well-balanced and entropy consistent numerical scheme for one-dimensional shallow water equations

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.