Numerical investigation to the full URE equations via predictor-corrector scheme

We produce the modified Rusanov (mR) method for the full ultra-relativistic Euler (URE) model. This system describes the flow of an ideal fluid of pressure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="sans-serif">p</mml:mi> </mml:math> , spatial part <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="sans-serif">u</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> of the dimensionless four-velocity and particle density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="sans-serif">n</mml:mi> </mml:math> . The nonlinear system in question is purely hyperbolic. The mR method is formulated as predictor and corrector phases. The predictor is based on the parameter of control <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , which is accountable for this scheme’s numerical diffusion. The balance conservation equation is reinstated by the corrector phase. Without utilizing Riemann problem solvers, the approach has the ability to compute the numerical flow that’s closest to the solution’s actually state. In the numerical simulation the results acquired via the mR method are compared with the Rusanov, the Lax-Friedrichs methods and reference solution with 10000 grid points calculated employing classical Rusanov. This numerical study confirms the effectiveness of the mR scheme. The mR technique can also be implemented to solve numerous other applied science models.