The difference variational bicomplex, which is the natural setting for systems of difference equations, is constructed and used to examine the geometric and algebraic properties of various systems. Exactness of the bicomplex gives a coordinate-free setting for finite difference variational problems, Euler--Lagrange equations and Noether's theorem. We also examine the connection between the condition for existence of a Hamiltonian and the multisymplecticity of systems of partial difference equations. Furthermore, we define difference multimomentum maps of multisymplectic systems, which yield their conservation laws. To conclude, we demonstrate how multisymplectic integrators can be comprehended even on non-uniform meshes through a generalized difference variational bicomplex.

翻译：差分变分双复形是差分方程系统的自然框架，本文构建了该复形并用于研究多种系统的几何与代数性质。双复形的正合性为有限差分变分问题、欧拉-拉格朗日方程及诺特定理提供了无坐标的设定。我们还探讨了哈密顿量存在条件与偏差分方程系统多辛性之间的关联。进一步，定义多辛系统的差分离散多重动量映射，由此导出其守恒律。最后，通过广义差分变分双复形，论证了即使在非均匀网格上也能理解多辛积分器的构造。