How did Mammoths survive in the tundra?

wikipedia:

Mammoths alive in the Arctic during the Last Glacial Maximum consumed mainly forbs, such as Artemisia; graminoids (grasses) were only a minor part of their diet.

The arctic tundra and steppe where the mammoths lived appears to have been dominated by forbs, not grass. There were richer in protein and easier to digest than grasses and wooden plants, which came to dominate the areas when the climate became wetter and warmer. This could have been a major contributor to why the arctic megafauna went extinct.

A forb or phorb is a herbaceous flowering plant that is not a graminoid (grass, sedge, or rush). The term is used in biology and in vegetation ecology, especially in relation to grasslands and understory.

In botany and ecology, graminoid refers to a herbaceous plant with a grass-like morphology, i.e. elongated culms with long, blade-like leaves. They are contrasted to forbs, herbaceous plants without grass-like features. The plants most often referred to include the families Poaceae (grasses in the strict sense), Cyperaceae (sedges), and Juncaceae (rushes). These are not closely related but belong to different clades in the order Poales. The grasses (Poaceae) are by far the largest family with some 12,000 species.

Perhaps they hibernated?

There were dozens of little flowering plants during the ice age.  Conditions changed.

I'm no biologist or zoologist, but I would assume they foraged for food wherever possible and used fat storages to provide them with energy during times of famine