Solving One-Dimensional Euler Equations Using Finite Volume Method with Second-Order MUSCL Upwind Scheme

The finite volume method for solving one-dimensional Euler equations is a widely used numerical approach in computational fluid dynamics (CFD), particularly suitable for handling discontinuous problems involving shocks. The MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws) format is a second-order upwind scheme that enhances computational accuracy while effectively suppressing numerical oscillations. Below is the computational procedure and key concepts for one-dimensional isentropic nozzle flow: ### Problem Description The one-dimensional Euler equations describe gas flow conservation laws (mass, momentum, energy conservation) within a nozzle. The nozzle contains an expansion section where a normal shock wave develops at a certain location. We need to solve this flow problem using the finite volume method combined with the MUSCL scheme. ### Computational Procedure Grid Generation Divide the nozzle into multiple control volumes (finite volume elements), with each cell storing conservative variables (density, momentum, energy). In code implementation, this typically involves creating arrays for primitive variables and flux calculations at cell interfaces. MUSCL Reconstruction The core of the MUSCL scheme lies in higher-order reconstruction of variables at cell interfaces to improve spatial discretization accuracy. Typically employing linear reconstruction: At cell boundaries, slope limiters (such as minmod or van Leer limiters) are applied to left and right states to prevent numerical oscillations. The upwind philosophy is maintained to ensure numerical stability. Code implementation often involves calculating limited slopes using minmod function: slope = minmod((u_i - u_{i-1})/Δx, (u_{i+1} - u_i)/Δx). Flux Calculation Employ Riemann solvers (such as Roe, HLLC) to compute numerical fluxes at cell interfaces. The reconstructed left and right states from MUSCL serve as inputs to the Riemann problem, enhancing flux calculation accuracy. The Roe solver implementation typically involves characteristic decomposition and flux difference splitting. Time Advancement Use explicit time integration methods (like Runge-Kutta schemes) for temporal advancement, ensuring second-order accuracy in time discretization. A typical implementation uses RK2 method: u^{n+1} = u^n + Δt/2*(k1 + k2) where k1 and k2 are intermediate flux evaluations. Shock Handling In the nozzle expansion section, shock appearance causes solution discontinuities. The MUSCL scheme automatically adjusts reconstruction slopes through limiters, degenerating to first-order format near shocks to avoid unphysical oscillations while maintaining second-order accuracy in smooth regions. The limiter function implementation ensures TVD property by bounding the reconstruction slopes. ### Key Points Second-order accuracy: Compared to first-order upwind schemes, the MUSCL format more accurately simulates flow details such as shock location and intensity. Limiter function: Ensures the scheme maintains TVD (Total Variation Diminishing) property near shocks, preventing numerical oscillations. Stability: The upwind characteristic combined with appropriate CFL conditions guarantees computational stability. Typical CFL implementation: Δt = CFL * Δx / max(|u| + c) where c is sound speed. Through this procedure, one can effectively simulate shock phenomena in nozzle flows and understand the application of MUSCL scheme in shock capturing and high-precision computations.