A well-balanced high-order scheme for shallow water equations with variable topography and temperature gradient is constructed. This scheme is of van Leer-type and is based on exact Riemann solvers. The scheme is shown to be able to capture almost exactly the stationary smooth solutions as well as stationary elementary discontinuities. Numerical tests show that the scheme gives a much better accuracy than the Godunov-type scheme and can work well even in the resonant regime. Wave interaction problems are also tested where the scheme possesses a good accuracy. It turns out that the superbee limiter can provide us with more accurate approximations than van Leer’s limiter.

构造了具有可变地形和温度梯度的浅水方程组的均衡高阶方案。该方案是van Leer型的，并且基于精确的Riemann求解器。结果表明，该方案几乎可以捕获平稳的平稳解以及平稳的基本不连续点。数值测试表明，该方案比Godunov型方案具有更高的精度，即使在共振状态下也能很好地工作。在该方案具有良好准确性的情况下，还测试了波浪相互作用问题。事实证明，与van Leer的限制器相比，超级蜜蜂限制器可以为我们提供更精确的近似值。