Runge-Kutta Discontinuous Galerkin Method Based on Flux Vector Splitting with Constrained Optimization-based TVB(D)-minmod Limiter for Solving Hyperbolic Conservation Laws

TL;DR

This paper develops a Runge-Kutta DG framework that integrates flux vector splitting (FVS) into the spatial discretization of hyperbolic conservation laws. A novel IS-$L^2$-TVB(D)-minmod limiter, formed as a constrained bi-objective optimization incorporating a smoothness factor from WENO and an $L^2$-error term, balances oscillation suppression with high-order accuracy; it is extended to two dimensions and combined with local characteristic reconstruction via interpolation-based transformation and Roe-based freezing. The method is validated on 1D/2D Euler and shallow water systems, showing optimal spatial convergence and robust shock-capturing capabilities across linear and nonlinear test problems, including standard Riemann problems. The work thereby provides a versatile, high-order DG approach with reduced oscillations, enhanced stability, and applicability to complex hyperbolic systems. Overall, the IS-$L^2$-TVB(D)-minmod limiter and FVS-DG flux framework offer a practical pathway to accurate, stable simulations of hyperbolic conservation laws in multiple dimensions.

Abstract

The flux vector splitting (FVS) method has firstly been incorporated into the discontinuous Galerkin (DG) framework for reconstructing the numerical fluxes required for the spatial semi-discrete formulation, setting it apart from the conventional DG approaches that typically utilize the Lax-Friedrichs flux scheme or classical Riemann solvers. The control equations of hyperbolic conservation systems are initially reformulated into a flux-split form. Subsequently, a variational approach is applied to this flux-split form, from which a DG spatial semi-discrete scheme based on FVS is derived. In order to suppress numerical pseudo-oscillations, the smoothness measurement function IS from the WENO limiter is integrated into the TVB(D)-minmod limiter, constructing an optimization problem based on the smoothness factor constraint, thereby realizing a TVB(D)-minmod limiter applicable to arbitrary high-order polynomial approximation. Subsequently, drawing on the ``reconstructed polynomial and the original high-order scheme's L2 -error constraint'' from the literature [1] , combined with our smoothness factor constraint, a bi-objective optimization problem is formulated to enable the TVB(D)-minmod limiter to balance oscillation suppression and high precision. As for hyperbolic conservation systems, limiters are typically required to be used in conjunction with local characteristic decomposition. To transform polynomials from the physical space to the characteristic space, an interpolation-based characteristic transformation scheme has been proposed, and its equivalence with the original moment characteristic transformation has been demonstrated in one-dimensional scenarios. Finally, the concept of ``flux vector splitting based on Jacobian eigenvalue decomposition'' has been applied to the conservative linear scalar transport equations and the nonlinear Burgers' equation.

Figures (19)

• Figure 1: 1D-Burgers' equation $u_t + (\frac{1}{2}u^2)_x = 0$ with initial condition $u_0(x)=\sin(x)$. The simulation is performed up to time $t=2.0$. $P^3$-polynomial approximations and uniquely spaced 32 cells. TVB parameter $M=1$. The $v$ for KXRCF indicator in scalar case is taken as $u_{i\pm1/2}$ from inside the cell $I_i$: (a) classical TVB-minmod limiter without discontinuity indicator; (b) classical TVB-minmod limiter with TVB discontinuity indicator; (c) classical TVB-minmod limiter with KXRCF discontinuity indicator
• Figure 2: 1D-Burgers' equation $u_t + (\frac{1}{2}u^2)_x = 0$ with initial condition $u_0(x)=\sin(x)$. The simulation is performed up to $t=2.0$. $P^3$ and $P^5$-polynomial approximations and uniquely spaced cells. Numerical results based on different limiters including IS-TVB-minmod limiter, $L^2$-TVB-minmod limiter, SimpleWENO and OEDG are compared with each other. Localized magnification has been applied to all sub-figures.
• Figure 3: 1D-Buckley-Leverett problem $u_t + (\frac{4u^2}{4u^2+(1-u)^2})_x = 0$ with initial condition $u=1$ when $-\frac{1}{2} \leq x \leq 0$ and $u=0$ elsewhere. The simulation is performed up to time $t=0.4$. $P^3$-polynomial approximation and uniquely spaced 80 cells. Numerical results based on different limiters including IS-TVB-minmod limiter, $L^2$-TVB-minmod limiter, WENO5-JS, SimpleWENO and OEDG are compared with each other.
• Figure 4: 2D-Burgers' problem $U_t + (\frac{1}{2}U)_x + (\frac{1}{2}U)_y = 0$ with smooth initial conditions $U_0(x,y)=\sin\left(\frac{\pi}{2}(x+y)\right)$ while a shock during the evolution. The simulation is performed up to time $t=\frac{1.5}{\pi}$ when a shock has already appeared. $P^3$-polynomial approximation and uniquely spaced rectangular 50 $\times$ 50 cells. Discontinuity is captured based on IS-TVB-minmod limiter ($\omega_{IS}=1,\ \omega_{L^2}=0$).
• Figure 5: 2D-Burgers' problem with discontinuous initial conditions. $U_t + (\frac{1}{2}U)_x + (\frac{1}{2}U)_y = 0$. At $t=0$, $U_0(x,y)=0.5$ when $x<0.05 \text{\ and\ } y<0.05$, $U_0(x,y)=0.8$ when $x>0.05 \text{\ and\ } y<0.05$, $U_0(x,y)=-1$ when $x>0.05 \text{\ and\ } y>0.05$, $U_0(x,y)=-0.2$ when $x<0.05 \text{\ and\ } y>0.05$. The simulation is performed up to time $t=0.05$. $P^3$-polynomial approximation and uniform rectangular 50 $\times$ 50 cells. Discontinuity is dealed with IS-TVB-minmod limiter ($\omega_{IS}=1,\ \omega_{L^2}=0$).

Theorems & Definitions (45)

• Definition 2.1: Normative Orthogonal System
• Definition 2.2: DG Weak Solution Space
• Definition 2.3: DG Weak Solution
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Definition 3.1: Homogeneous Function
• Remark 3.3
• Remark 3.4
• Remark 3.5