We present a numerical method to obtain self-similar solutions of the ideal magnetohydrodynamics (MHD) equations. Under a self-similar transformation, the initial value problem (IVP) is converted into a boundary value problem (BVP) by eliminating time and transforming the system to self-similar coordinates $(\xi \equiv x/t,\eta \equiv y/t)$. The ideal MHD system of equations is augmented by a generalized Lagrange multiplier (GLM) to maintain the solenoidal condition on the magnetic field. The self-similar solution to the BVP is solved using an iterative method, and implemented using the p4est adaptive mesh refinement (AMR) framework. Existing Riemann solvers (e.g., Roe, HLLD etc.) can be modified in a relatively straightforward manner and used in the present method. Numerical tests illustrate that the present self-similar solution to the BVP exhibits sharper discontinuities than the corresponding one solved by the IVP. We compare and contrast the IVP and BVP solutions in several one dimensional shock-tube test problems and two dimensional test cases include shock wave refraction at a contact discontinuity, reflection at a solid wall, and shock wave diffraction over a right angle corner.

我们提出了一种数值方法来获得理想磁流体动力学 (MHD) 方程的自相似解。在自相似变换下，通过消除时间并将系统变换为自相似坐标，将初值问题（IVP）转化为边值问题（BVP）$(\xi \equiv X/吨,\eta \equiv 是/吨)$. 理想的 MHD 方程组由广义拉格朗日乘子 (GLM) 增强，以保持磁场的螺线管条件。BVP 的自相似解使用迭代方法求解，并使用 p4est 自适应网格细化 (AMR) 框架实现。现有的黎曼求解器（例如，Roe、HLLD 等）可以以相对直接的方式修改并用于本方法中。数值测试表明，目前 BVP 的自相似解比 IVP 解出的相应解表现出更尖锐的不连续性。我们在几个一维冲击管测试问题和二维测试案例中比较和对比了 IVP 和 BVP 解决方案，包括接触不连续处的冲击波折射、实体壁处的反射、